Chapter 9: Rational Numbers

Exercise: 9.1 (10)

 

Question: 1

List five rational numbers between:

(i)             1 and 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaGimaaaa@3C3C@

(ii)          2 and1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaabccacaqGHbGaaeOBaiaabsgacqGHsislcaaIXaaaaa@3C88@

(iii)      4 5  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3DDD@

(iv)       1 2  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGOmaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3DD7@

Solution

Rational number is any number that can be expressed in the form of p/q, where p and q are integers and q 0.

(i)             We need to find five rational number between 1 and 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaGimaaaa@3C3C@

So, we can write 1 and 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaGimaaaa@3C3C@  as rational numbers with denominator 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaaaa@3698@ .

   1= 6 6  and 0= 0 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaeyOeI0IaaGymaiabg2da9maalaaabaGaeyOe I0IaaGOnaaqaaiaaiAdaaaGaaeiiaiaabggacaqGUbGaaeizaiaabc cacaaIWaGaeyypa0ZaaSaaaeaacaaIWaaabaGaaGOnaaaaaaa@4695@

     6 6 < 5 6 < 4 6 < 3 6 < 2 6 < 1 6 <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaGOnaaqaaiaa iAdaaaGaeyipaWZaaSaaaeaacqGHsislcaaI1aaabaGaaGOnaaaacq GH8aapdaWcaaqaaiabgkHiTiaaisdaaeaacaaI2aaaaiabgYda8maa laaabaGaeyOeI0IaaG4maaqaaiaaiAdaaaGaeyipaWZaaSaaaeaacq GHsislcaaIYaaabaGaaGOnaaaacqGH8aapdaWcaaqaaiabgkHiTiaa igdaaeaacaaI2aaaaiabgYda8iaaicdaaaa@4F53@

   1< 5 6 < 2 3 < 1 2 < 1 3 < 1 6 <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaeyOeI0IaaGymaiabgYda8maalaaabaGaeyOe I0IaaGynaaqaaiaaiAdaaaGaeyipaWZaaSaaaeaacqGHsislcaaIYa aabaGaaG4maaaacqGH8aapdaWcaaqaaiabgkHiTiaaigdaaeaacaaI YaaaaiabgYda8maalaaabaGaeyOeI0IaaGymaaqaaiaaiodaaaGaey ipaWZaaSaaaeaacqGHsislcaaIXaaabaGaaGOnaaaacqGH8aapcaaI Waaaaa@4EEB@

Therefore, five rational numbers between 1 and 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaaGimaaaa@3C3C@  would be

5 6 , 2 3 , 1 2 , 1 3 , 1 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGOnaaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaG4maaaacaGGSaWaaSaaaeaacqGHsislcaaIXaaabaGaaG OmaaaacaGGSaWaaSaaaeaacqGHsislcaaIXaaabaGaaG4maaaacaGG SaWaaSaaaeaacqGHsislcaaIXaaabaGaaGOnaaaaaaa@44EB@

(ii)          We need to find five rational number between 2 and1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaabccacaqGHbGaaeOBaiaabsgacqGHsislcaaIXaaaaa@3C88@

Let us write 2 and1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaabccacaqGHbGaaeOBaiaabsgacaaMe8UaeyOeI0IaaGymaaaa @3E15@  as rational numbers with denominator 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374D@ .

   2= 20 10  and1= 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaeyOeI0IaaGOmaiabg2da9maalaaabaGaeyOe I0IaaGOmaiaaicdaaeaacaaIXaGaaGimaaaacaqGGaGaaeyyaiaab6 gacaqGKbGaaGjbVlabgkHiTiaaigdacqGH9aqpdaWcaaqaaiabgkHi TiaaigdacaaIWaaabaGaaGymaiaaicdaaaaaaa@4C36@

     20 10 < 19 10 < 18 10 < 17 10 < 16 10 < 15 10 < 10 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaGOmaiaaicda aeaacaaIXaGaaGimaaaacqGH8aapdaWcaaqaaiabgkHiTiaaigdaca aI5aaabaGaaGymaiaaicdaaaGaeyipaWZaaSaaaeaacqGHsislcaaI XaGaaGioaaqaaiaaigdacaaIWaaaaiabgYda8maalaaabaGaeyOeI0 IaaGymaiaaiEdaaeaacaaIXaGaaGimaaaacqGH8aapdaWcaaqaaiab gkHiTiaaigdacaaI2aaabaGaaGymaiaaicdaaaGaeyipaWZaaSaaae aacqGHsislcaaIXaGaaGynaaqaaiaaigdacaaIWaaaaiabgYda8maa laaabaGaeyOeI0IaaGymaiaaicdaaeaacaaIXaGaaGimaaaaaaa@5B2F@

   2< 19 10 < 9 5 < 17 10 < 8 5 < 3 2 <1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaeyOeI0IaaGOmaiabgYda8maalaaabaGaeyOe I0IaaGymaiaaiMdaaeaacaaIXaGaaGimaaaacqGH8aapdaWcaaqaai abgkHiTiaaiMdaaeaacaaI1aaaaiabgYda8maalaaabaGaeyOeI0Ia aGymaiaaiEdaaeaacaaIXaGaaGimaaaacqGH8aapdaWcaaqaaiabgk HiTiaaiIdaaeaacaaI1aaaaiabgYda8maalaaabaGaeyOeI0IaaG4m aaqaaiaaikdaaaGaeyipaWJaeyOeI0IaaGymaaaa@52D8@

Therefore, five rational numbers between 2 and1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaabccacaqGHbGaaeOBaiaabsgacaaMe8UaeyOeI0IaaGymaaaa @3E15@  would be

19 10 , 9 5 , 17 10 , 8 5 , 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaGaaGyoaaqaaiaaigdacaaIWaaaaiaacYcadaWcaaqa aiabgkHiTiaaiMdaaeaacaaI1aaaaiaacYcadaWcaaqaaiabgkHiTi aaigdacaaI3aaabaGaaGymaiaaicdaaaGaaiilamaalaaabaGaeyOe I0IaaGioaaqaaiaaiwdaaaGaaiilamaalaaabaGaeyOeI0IaaG4maa qaaiaaikdaaaaaaa@47E9@

(iii)      4 5  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGOmaaqaaiaaiodaaaaaaa@3ECA@

Let us write 4 5  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGOmaaqaaiaaiodaaaaaaa@3ECA@  as rational numbers with the same denominators.

    4 5 = 36 45  and  2 3 = 30 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaI0aaabaGaaGynaaaa cqGH9aqpdaWcaaqaaiabgkHiTiaaiodacaaI2aaabaGaaGinaiaaiw daaaGaaeiiaiaabggacaqGUbGaaeizaiaabccadaWcaaqaaiabgkHi TiaaikdaaeaacaaIZaaaaiabg2da9maalaaabaGaeyOeI0IaaG4mai aaicdaaeaacaaI0aGaaGynaaaaaaa@4D04@

     36 45 < 35 45 < 34 45 < 33 45 < 32 45 < 31 45 < 30 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaG4maiaaiAda aeaacaaI0aGaaGynaaaacqGH8aapdaWcaaqaaiabgkHiTiaaiodaca aI1aaabaGaaGinaiaaiwdaaaGaeyipaWZaaSaaaeaacqGHsislcaaI ZaGaaGinaaqaaiaaisdacaaI1aaaaiabgYda8maalaaabaGaeyOeI0 IaaG4maiaaiodaaeaacaaI0aGaaGynaaaacqGH8aapdaWcaaqaaiab gkHiTiaaiodacaaIYaaabaGaaGinaiaaiwdaaaGaeyipaWZaaSaaae aacqGHsislcaaIZaGaaGymaaqaaiaaisdacaaI1aaaaiabgYda8maa laaabaGaeyOeI0IaaG4maiaaicdaaeaacaaI0aGaaGynaaaaaaa@5B66@

    4 5 < 7 9 < 34 45 < 11 15 < 32 45 < 31 45 < 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaI0aaabaGaaGynaaaa cqGH8aapdaWcaaqaaiabgkHiTiaaiEdaaeaacaaI5aaaaiabgYda8m aalaaabaGaeyOeI0IaaG4maiaaisdaaeaacaaI0aGaaGynaaaacqGH 8aapdaWcaaqaaiabgkHiTiaaigdacaaIXaaabaGaaGymaiaaiwdaaa GaeyipaWZaaSaaaeaacqGHsislcaaIZaGaaGOmaaqaaiaaisdacaaI 1aaaaiabgYda8maalaaabaGaeyOeI0IaaG4maiaaigdaaeaacaaI0a GaaGynaaaacqGH8aapdaWcaaqaaiabgkHiTiaaikdaaeaacaaIZaaa aaaa@576E@

Therefore, five rational numbers between 4 5  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGOmaaqaaiaaiodaaaaaaa@3ECA@  would be

7 9 , 34 45 , 11 15 , 32 45 , 31 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGyoaaaacaGGSaWaaSaaaeaacqGHsislcaaI ZaGaaGinaaqaaiaaisdacaaI1aaaaiaacYcadaWcaaqaaiabgkHiTi aaigdacaaIXaaabaGaaGymaiaaiwdaaaGaaiilamaalaaabaGaeyOe I0IaaG4maiaaikdaaeaacaaI0aGaaGynaaaacaGGSaWaaSaaaeaacq GHsislcaaIZaGaaGymaaqaaiaaisdacaaI1aaaaaaa@4AE0@

(iv)       1 2  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGOmaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3DD7@

Let us write 1 2  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGOmaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3DD7@  as rational numbers with the same denominators.

    1 2 = 3 6  and  2 3 = 4 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIXaaabaGaaGOmaaaa cqGH9aqpdaWcaaqaaiabgkHiTiaaiodaaeaacaaI2aaaaiaabccaca qGHbGaaeOBaiaabsgacaqGGaWaaSaaaeaacaaIYaaabaGaaG4maaaa cqGH9aqpdaWcaaqaaiaaisdaaeaacaaI2aaaaaaa@4831@

     3 6 < 2 6 < 1 6 <0< 1 6 < 2 6 < 3 6 < 4 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaG4maaqaaiaa iAdaaaGaeyipaWZaaSaaaeaacqGHsislcaaIYaaabaGaaGOnaaaacq GH8aapdaWcaaqaaiabgkHiTiaaigdaaeaacaaI2aaaaiabgYda8iaa icdacqGH8aapdaWcaaqaaiaaigdaaeaacaaI2aaaaiabgYda8maala aabaGaaGOmaaqaaiaaiAdaaaGaeyipaWZaaSaaaeaacaaIZaaabaGa aGOnaaaacqGH8aapdaWcaaqaaiaaisdaaeaacaaI2aaaaaaa@4F15@

    1 2 < 1 3 < 1 6 <0< 1 6 < 1 3 < 1 2 < 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIXaaabaGaaGOmaaaa cqGH8aapdaWcaaqaaiabgkHiTiaaigdaaeaacaaIZaaaaiabgYda8m aalaaabaGaeyOeI0IaaGymaaqaaiaaiAdaaaGaeyipaWJaaGimaiab gYda8maalaaabaGaaGymaaqaaiaaiAdaaaGaeyipaWZaaSaaaeaaca aIXaaabaGaaG4maaaacqGH8aapdaWcaaqaaiaaigdaaeaacaaIYaaa aiabgYda8maalaaabaGaaGOmaaqaaiaaiodaaaaaaa@4F78@

Therefore, five rational numbers between 1 2  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGOmaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3DD7@  would be

1 3 , 1 6 ,0, 1 6 , 1 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaG4maaaacaGGSaWaaSaaaeaacqGHsislcaaI XaaabaGaaGOnaaaacaGGSaGaaGimaiaacYcadaWcaaqaaiaaigdaae aacaaI2aaaaiaacYcadaWcaaqaaiaaigdaaeaacaaIZaaaaiaac6ca aaa@4204@

Question: 2

Write four more rational numbers in each of the following patterns:

(i)             3 5 , 6 10 , 9 15 , 12 20 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaI 2aaabaGaaGymaiaaicdaaaGaaiilamaalaaabaGaeyOeI0IaaGyoaa qaaiaaigdacaaI1aaaaiaacYcadaWcaaqaaiabgkHiTiaaigdacaaI YaaabaGaaGOmaiaaicdaaaGaaiilaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@4BA9@

(ii)          1 4 , 2 8 , 3 12 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGinaaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaGioaaaacaGGSaWaaSaaaeaacqGHsislcaaIZaaabaGaaG ymaiaaikdaaaGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaGGUaGaaiOlaaaa@464C@

(iii)      1 6 , 2 12 , 3 18 , 4 24 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGOnaaaacaGGSaWaaSaaaeaacaaIYaaabaGa eyOeI0IaaGymaiaaikdaaaGaaiilamaalaaabaGaaG4maaqaaiabgk HiTiaaigdacaaI4aaaaiaacYcadaWcaaqaaiaaisdaaeaacqGHsisl caaIYaGaaGinaaaacaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaGaaiOlaiaac6cacaGGUaaaaa@4AEE@

(iv)       2 3 , 2 3 , 4 6 , 6 9 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4maaaacaGGSaWaaSaaaeaacaaIYaaabaGa eyOeI0IaaG4maaaacaGGSaWaaSaaaeaacaaI0aaabaGaeyOeI0IaaG OnaaaacaGGSaWaaSaaaeaacaaI2aaabaGaeyOeI0IaaGyoaaaacaGG SaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaaaaa@48C1@

Solution

(i)             3 5 , 6 10 , 9 15 , 12 20 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaI 2aaabaGaaGymaiaaicdaaaGaaiilamaalaaabaGaeyOeI0IaaGyoaa qaaiaaigdacaaI1aaaaiaacYcadaWcaaqaaiabgkHiTiaaigdacaaI YaaabaGaaGOmaiaaicdaaaGaaiilaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaaaa@4BA9@

    3×1 5×1 , 3×2 5×2 , 3×3 5×3 , 3×4 5×4 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIZaGaey41aqRaaGym aaqaaiaaiwdacqGHxdaTcaaIXaaaaiaacYcadaWcaaqaaiabgkHiTi aaiodacqGHxdaTcaaIYaaabaGaaGynaiabgEna0kaaikdaaaGaaiil amaalaaabaGaeyOeI0IaaG4maiabgEna0kaaiodaaeaacaaI1aGaey 41aqRaaG4maaaacaGGSaWaaSaaaeaacqGHsislcaaIZaGaey41aqRa aGinaaqaaiaaiwdacqGHxdaTcaaI0aaaaiaacYcacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa@63A0@

We observe that numerator is multiple of 3 and denominator is multiple of 5.

Therefore, the next four rational numbers of this pattern would be

3×5 5×5 , 3×6 5×6 , 3×7 5×7 , 3×8 5×8 = 15 25 , 18 30 , 21 35 , 24 40 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaaiodacqGHxdaTcaaI1aaabaGaaGynaiabgEna0kaa iwdaaaGaaiilamaalaaabaGaeyOeI0IaaG4maiabgEna0kaaiAdaae aacaaI1aGaey41aqRaaGOnaaaacaGGSaWaaSaaaeaacqGHsislcaaI ZaGaey41aqRaaG4naaqaaiaaiwdacqGHxdaTcaaI3aaaaiaacYcada WcaaqaaiabgkHiTiaaiodacqGHxdaTcaaI4aaabaGaaGynaiabgEna 0kaaiIdaaaaabaGaeyypa0ZaaSaaaeaacqGHsislcaaIXaGaaGynaa qaaiaaikdacaaI1aaaaiaacYcadaWcaaqaaiabgkHiTiaaigdacaaI 4aaabaGaaG4maiaaicdaaaGaaiilamaalaaabaGaeyOeI0IaaGOmai aaigdaaeaacaaIZaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaI YaGaaGinaaqaaiaaisdacaaIWaaaaaaaaa@6B67@

(ii)          1 4 , 2 8 , 3 12 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGinaaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaGioaaaacaGGSaWaaSaaaeaacqGHsislcaaIZaaabaGaaG ymaiaaikdaaaGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGa aiOlaiaac6cacaGGUaGaaiOlaaaa@464C@

    1×1 4×1 , 1×2 4×2 , 1×3 4×3 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIXaGaey41aqRaaGym aaqaaiaaisdacqGHxdaTcaaIXaaaaiaacYcadaWcaaqaaiabgkHiTi aaigdacqGHxdaTcaaIYaaabaGaaGinaiabgEna0kaaikdaaaGaaiil amaalaaabaGaeyOeI0IaaGymaiabgEna0kaaiodaaeaacaaI0aGaey 41aqRaaG4maaaacaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6ca caGGUaGaaiOlaiaac6cacaGGUaaaaa@5AC4@

Therefore, the next four rational numbers of this pattern would be

1×4 4×4 , 1×5 4×5 , 1×6 4×6 , 1×7 4×7 = 4 16 , 5 20 , 6 24 , 7 28 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaaigdacqGHxdaTcaaI0aaabaGaaGinaiabgEna0kaa isdaaaGaaiilamaalaaabaGaeyOeI0IaaGymaiabgEna0kaaiwdaae aacaaI0aGaey41aqRaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaI XaGaey41aqRaaGOnaaqaaiaaisdacqGHxdaTcaaI2aaaaiaacYcada WcaaqaaiabgkHiTiaaigdacqGHxdaTcaaI3aaabaGaaGinaiabgEna 0kaaiEdaaaaabaGaeyypa0ZaaSaaaeaacqGHsislcaaI0aaabaGaaG ymaiaaiAdaaaGaaiilamaalaaabaGaeyOeI0IaaGynaaqaaiaaikda caaIWaaaaiaacYcadaWcaaqaaiabgkHiTiaaiAdaaeaacaaIYaGaaG inaaaacaGGSaWaaSaaaeaacqGHsislcaaI3aaabaGaaGOmaiaaiIda aaaaaaa@686C@

(iii)      1 6 , 2 12 , 3 18 , 4 24 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGOnaaaacaGGSaWaaSaaaeaacaaIYaaabaGa eyOeI0IaaGymaiaaikdaaaGaaiilamaalaaabaGaaG4maaqaaiabgk HiTiaaigdacaaI4aaaaiaacYcadaWcaaqaaiaaisdaaeaacqGHsisl caaIYaGaaGinaaaacaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaGaaiOlaiaac6cacaGGUaaaaa@4AEE@

    1×1 6×1 , 1×2 6×2 , 1×3 6×3 , 1×4 6×4 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIXaGaey41aqRaaGym aaqaaiaaiAdacqGHxdaTcaaIXaaaaiaacYcadaWcaaqaaiaaigdacq GHxdaTcaaIYaaabaGaeyOeI0IaaGOnaiabgEna0kaaikdaaaGaaiil amaalaaabaGaaGymaiabgEna0kaaiodaaeaacqGHsislcaaI2aGaey 41aqRaaG4maaaacaGGSaWaaSaaaeaacaaIXaGaey41aqRaaGinaaqa aiabgkHiTiaaiAdacqGHxdaTcaaI0aaaaiaacYcacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa@639C@

Therefore, the next four rational numbers of this pattern would be

1×5 6×5 , 1×6 6×6 , 1×7 6×7 , 1×8 6×8 = 5 30 , 6 36 , 7 42 , 8 48 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaaigdacqGHxdaTcaaI1aaabaGaeyOeI0IaaGOnaiabgEna0kaa iwdaaaGaaiilamaalaaabaGaaGymaiabgEna0kaaiAdaaeaacqGHsi slcaaI2aGaey41aqRaaGOnaaaacaGGSaWaaSaaaeaacaaIXaGaey41 aqRaaG4naaqaaiabgkHiTiaaiAdacqGHxdaTcaaI3aaaaiaacYcada WcaaqaaiaaigdacqGHxdaTcaaI4aaabaGaeyOeI0IaaGOnaiabgEna 0kaaiIdaaaaabaGaeyypa0ZaaSaaaeaacaaI1aaabaGaeyOeI0IaaG 4maiaaicdaaaGaaiilamaalaaabaGaaGOnaaqaaiabgkHiTiaaioda caaI2aaaaiaacYcadaWcaaqaaiaaiEdaaeaacqGHsislcaaI0aGaaG OmaaaacaGGSaWaaSaaaeaacaaI4aaabaGaeyOeI0IaaGinaiaaiIda aaaaaaa@6885@

(iv)       2 3 , 2 3 , 4 6 , 6 9 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4maaaacaGGSaWaaSaaaeaacaaIYaaabaGa eyOeI0IaaG4maaaacaGGSaWaaSaaaeaacaaI0aaabaGaeyOeI0IaaG OnaaaacaGGSaWaaSaaaeaacaaI2aaabaGaeyOeI0IaaGyoaaaacaGG SaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6 cacaGGUaaaaa@48C1@

    2×1 3×1 , 2×1 3×1 , 2×2 3×2 , 2×3 3×3 ,......... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIYaGaey41aqRaaGym aaqaaiaaiodacqGHxdaTcaaIXaaaaiaacYcadaWcaaqaaiabgkHiTi aaikdacqGHxdaTcqGHsislcaaIXaaabaGaaG4maiabgEna0kabgkHi TiaaigdaaaGaaiilamaalaaabaGaeyOeI0IaaGOmaiabgEna0kabgk HiTiaaikdaaeaacaaIZaGaey41aqRaeyOeI0IaaGOmaaaacaGGSaWa aSaaaeaacqGHsislcaaIYaGaey41aqRaeyOeI0IaaG4maaqaaiaaio dacqGHxdaTcqGHsislcaaIZaaaaiaacYcacaGGUaGaaiOlaiaac6ca caGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6caaaa@691C@

Therefore, the next four rational numbers of this pattern would be

2×4 3×4 , 2×5 3×5 , 2×6 3×6 , 2×7 3×7 = 8 12 , 10 15 , 12 18 , 14 21 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaaikdacqGHxdaTcqGHsislcaaI0aaabaGaaG4maiab gEna0kabgkHiTiaaisdaaaGaaiilamaalaaabaGaeyOeI0IaaGOmai abgEna0kabgkHiTiaaiwdaaeaacaaIZaGaey41aqRaeyOeI0IaaGyn aaaacaGGSaWaaSaaaeaacqGHsislcaaIYaGaey41aqRaeyOeI0IaaG OnaaqaaiaaiodacqGHxdaTcqGHsislcaaI2aaaaiaacYcadaWcaaqa aiabgkHiTiaaikdacqGHxdaTcqGHsislcaaI3aaabaGaaG4maiabgE na0kabgkHiTiaaiEdaaaaabaGaeyypa0ZaaSaaaeaacaaI4aaabaGa eyOeI0IaaGymaiaaikdaaaGaaiilamaalaaabaGaaGymaiaaicdaae aacqGHsislcaaIXaGaaGynaaaacaGGSaWaaSaaaeaacaaIXaGaaGOm aaqaaiabgkHiTiaaigdacaaI4aaaaiaacYcadaWcaaqaaiaaigdaca aI0aaabaGaeyOeI0IaaGOmaiaaigdaaaaaaaa@71F9@

Question: 3

Give four rational numbers equivalent to:

(i)             2 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4naaaaaaa@3852@

(ii)          5 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaeyOeI0IaaG4maaaaaaa@3851@

(iii)      4 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGyoaaaaaaa@3769@

Solution

(i)             2 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4naaaaaaa@3852@

Next four

2×2 7×2 = 4 14 , 2×3 7×3 = 6 21 , 2×4 7×4 = 8 28 , 2×5 7×5 = 10 35 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaikdacqGHxdaTcaaIYaaabaGaaG4naiabgEna0kaa ikdaaaGaeyypa0ZaaSaaaeaacqGHsislcaaI0aaabaGaaGymaiaais daaaGaaiilaaqaamaalaaabaGaeyOeI0IaaGOmaiabgEna0kaaioda aeaacaaI3aGaey41aqRaaG4maaaacqGH9aqpdaWcaaqaaiabgkHiTi aaiAdaaeaacaaIYaGaaGymaaaacaGGSaaabaWaaSaaaeaacqGHsisl caaIYaGaey41aqRaaGinaaqaaiaaiEdacqGHxdaTcaaI0aaaaiabg2 da9maalaaabaGaeyOeI0IaaGioaaqaaiaaikdacaaI4aaaaiaacYca aeaadaWcaaqaaiabgkHiTiaaikdacqGHxdaTcaaI1aaabaGaaG4nai abgEna0kaaiwdaaaGaeyypa0ZaaSaaaeaacqGHsislcaaIXaGaaGim aaqaaiaaiodacaaI1aaaaaaaaa@6A27@

Therefore, four equivalent rational numbers are 4 14 , 6 21 , 8 28 , 10 35 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGymaiaaisdaaaGaaiilamaalaaabaGaeyOe I0IaaGOnaaqaaiaaikdacaaIXaaaaiaacYcadaWcaaqaaiabgkHiTi aaiIdaaeaacaaIYaGaaGioaaaacaGGSaWaaSaaaeaacqGHsislcaaI XaGaaGimaaqaaiaaiodacaaI1aaaaiaac6caaaa@462D@

(ii)          5 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaeyOeI0IaaG4maaaaaaa@3851@

5×2 3×2 = 10 6 , 5×3 3×3 = 15 9 , 5×4 3×4 = 20 12 , 5×5 3×5 = 25 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiwdacqGHxdaTcaaIYaaabaGaeyOeI0IaaG4maiabgEna0kaa ikdaaaGaeyypa0ZaaSaaaeaacaaIXaGaaGimaaqaaiabgkHiTiaaiA daaaGaaiilaaqaamaalaaabaGaaGynaiabgEna0kaaiodaaeaacqGH sislcaaIZaGaey41aqRaaG4maaaacqGH9aqpdaWcaaqaaiaaigdaca aI1aaabaGaeyOeI0IaaGyoaaaacaGGSaaabaWaaSaaaeaacaaI1aGa ey41aqRaaGinaaqaaiabgkHiTiaaiodacqGHxdaTcaaI0aaaaiabg2 da9maalaaabaGaaGOmaiaaicdaaeaacqGHsislcaaIXaGaaGOmaaaa caGGSaaabaWaaSaaaeaacaaI1aGaey41aqRaaGynaaqaaiabgkHiTi aaiodacqGHxdaTcaaI1aaaaiabg2da9maalaaabaGaaGOmaiaaiwda aeaacqGHsislcaaIXaGaaGynaaaaaaaa@6AD8@

Therefore, four equivalent rational numbers are 10 6 , 15 9 , 20 12 , 25 15 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGimaaqaaiabgkHiTiaaiAdaaaGaaiilamaalaaabaGaaGym aiaaiwdaaeaacqGHsislcaaI5aaaaiaacYcadaWcaaqaaiaaikdaca aIWaaabaGaeyOeI0IaaGymaiaaikdaaaGaaiilamaalaaabaGaaGOm aiaaiwdaaeaacqGHsislcaaIXaGaaGynaaaacaGGUaaaaa@46E2@

(iii)      4 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGyoaaaaaaa@3769@

4×2 9×2 = 8 18 , 4×3 9×3 = 12 27 , 4×4 9×4 = 16 36 , 4×5 9×5 = 20 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaisdacqGHxdaTcaaIYaaabaGaaGyoaiabgEna0kaaikdaaaGa eyypa0ZaaSaaaeaacaaI4aaabaGaaGymaiaaiIdaaaGaaiilaaqaam aalaaabaGaaGinaiabgEna0kaaiodaaeaacaaI5aGaey41aqRaaG4m aaaacqGH9aqpdaWcaaqaaiaaigdacaaIYaaabaGaaGOmaiaaiEdaaa GaaiilaaqaamaalaaabaGaaGinaiabgEna0kaaisdaaeaacaaI5aGa ey41aqRaaGinaaaacqGH9aqpdaWcaaqaaiaaigdacaaI2aaabaGaaG 4maiaaiAdaaaGaaiilaaqaamaalaaabaGaaGinaiabgEna0kaaiwda aeaacaaI5aGaey41aqRaaGynaaaacqGH9aqpdaWcaaqaaiaaikdaca aIWaaabaGaaGinaiaaiwdaaaaaaaa@644E@

Therefore, four equivalent rational numbers are 8 18 , 12 27 , 16 36 , 20 45 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aaabaGaaGymaiaaiIdaaaGaaiilamaalaaabaGaaGymaiaaikda aeaacaaIYaGaaG4naaaacaGGSaWaaSaaaeaacaaIXaGaaGOnaaqaai aaiodacaaI2aaaaiaacYcadaWcaaqaaiaaikdacaaIWaaabaGaaGin aiaaiwdaaaGaaiOlaaaa@43F8@

Question: 4

Draw the number line and represent the following rational numbers on it:

(i)             3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaGinaaaaaaa@3763@

(ii)          5 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGioaaaaaaa@3856@

(iii)      7 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGinaaaaaaa@3854@

(iv)       7 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aaabaGaaGioaaaaaaa@376B@

Solution

(i)             3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaGinaaaaaaa@3763@

Given fraction represent 3 parts out of 4 equal parts. Hence, the space between the integers 0 and 1 on the number line must be divided into 4 parts.

(ii)          5 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGioaaaaaaa@3856@

Given fraction represent 5 parts out of 8 equal parts. Negative sign represents that it is on the negative side of the number line. Hence, the space between the integers -1 and 0 must be divided into 8 equal parts on the number line. 

(iii)      7 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGinaaaaaaa@3854@

We can write given fraction as

7 4 =1 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGinaaaacqGH9aqpcqGHsislcaaIXaWaaSaa aeaacaaIZaaabaGaaGinaaaaaaa@3C8D@

This fraction represents 1 full part and 3 part out of 4 equal parts. Therefore, each space between the integers 0 and -1 and also between -2 and -1 must be divided into 4 equal parts on the number line.

(iv)       7 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aaabaGaaGioaaaaaaa@376B@

 Given fraction represents 7 parts out of 8 equal parts. Therefore, space between the  integers 0 and 1 and also between 1 and 2 must be divided into 8 equal parts on the number line. 

Question: 5

The points P, Q, R, S, T, U, A and B on the number line are such that, TR = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DE@  RS = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DE@  SU and AP = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DE@  PQ = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DE@  QB. Name the rational numbers represented by P, Q, R and S.

Solution

In the given number line, each part which is between the two numbers is divided into 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3695@  parts.

Distance between U and T = 1 Unit

It is divided into 3 equal parts.

TR=RS=SU= 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeivaiaabk facaqG9aGaaeOuaiaabofacaqG9aGaaGzaVlaaykW7caqGtbGaaeyv aiabg2da9maalaaabaGaaGymaaqaaiaaiodaaaaaaa@4201@  

R=1 1 3 = 31 3 = 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaayk W7cqGH9aqpcaaMc8UaeyOeI0IaaGymaiabgkHiTmaalaaabaGaaGym aaqaaiaaiodaaaGaeyypa0ZaaSaaaeaacqGHsislcaaIZaGaeyOeI0 IaaGymaaqaaiaaiodaaaGaeyypa0ZaaSaaaeaacqGHsislcaaI0aaa baGaaG4maaaaaaa@478C@  

S=1 2 3 = 32 3 = 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaayk W7cqGH9aqpcaaMc8UaeyOeI0IaaGymaiabgkHiTmaalaaabaGaaGOm aaqaaiaaiodaaaGaeyypa0ZaaSaaaeaacqGHsislcaaIZaGaeyOeI0 IaaGOmaaqaaiaaiodaaaGaeyypa0ZaaSaaaeaacqGHsislcaaI1aaa baGaaG4maaaaaaa@4790@

Similarly,

AB = 1 unit, and it is divided into 3 equal parts.

Therefore,

P=2+ 1 3 = 6+1 3 = 7 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaayk W7cqGH9aqpcaaMc8UaaGOmaiabgUcaRmaalaaabaGaaGymaaqaaiaa iodaaaGaeyypa0ZaaSaaaeaacaaI2aGaey4kaSIaaGymaaqaaiaaio daaaGaeyypa0ZaaSaaaeaacaaI3aaabaGaaG4maaaaaaa@44B2@

Q=2+ 2 3 = 6+2 3 = 8 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyuaiaayk W7cqGH9aqpcaaMc8UaaGOmaiabgUcaRmaalaaabaGaaGOmaaqaaiaa iodaaaGaeyypa0ZaaSaaaeaacaaI2aGaey4kaSIaaGOmaaqaaiaaio daaaGaeyypa0ZaaSaaaeaacaaI4aaabaGaaG4maaaaaaa@44B6@

Thus, the rational numbers represented P, Q, R and S are 7 3 , 8 3 , 4 3  and  5 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aaabaGaaG4maaaacaGGSaWaaSaaaeaacaaI4aaabaGaaG4maaaa caGGSaWaaSaaaeaacqGHsislcaaI0aaabaGaaG4maaaacaqGGaGaae yyaiaab6gacaqGKbGaaeiiamaalaaabaGaeyOeI0IaaGynaaqaaiaa iodaaaaaaa@4348@  respectively.

Question: 6

Which of the following pairs represent the same rational number?

(i)             7 21  and  3 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGOmaiaaigdaaaGaaeiiaiaabggacaqGUbGa aeizaiaabccadaWcaaqaaiaaiodaaeaacaaI5aaaaaaa@3E9F@

(ii)          16 20  and  20 25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaGaaGOnaaqaaiaaikdacaaIWaaaaiaabccacaqGHbGa aeOBaiaabsgacaqGGaWaaSaaaeaacaaIYaGaaGimaaqaaiabgkHiTi aaikdacaaI1aaaaaaa@41B6@

(iii)      2 3  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaeyOeI0IaaG4maaaacaqGGaGaaeyyaiaab6ga caqGKbGaaeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3EC6@

(iv)       3 5  and  12 20 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGymaiaaikdaaeaacaaIYaGaaGimaa aaaaa@403D@

(v)          8 5  and  24 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aaabaGaeyOeI0IaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGOmaiaaisdaaeaacaaIXaGaaGynaa aaaaa@4049@

(vi)       1 3  and  1 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaG4maaaacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiamaa laaabaGaeyOeI0IaaGymaaqaaiaaiMdaaaaaaa@3DDD@

(vii)    5 9  and  5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaeyOeI0IaaGyoaaaacaqGGaGaaeyyaiaab6ga caqGKbGaaeiiamaalaaabaGaaGynaaqaaiabgkHiTiaaiMdaaaaaaa@3FC5@

Solution

(i)                         7 21  and  3 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGOmaiaaigdaaaGaaeiiaiaabggacaqGUbGa aeizaiaabccadaWcaaqaaiaaiodaaeaacaaI5aaaaaaa@3E9F@

  7 21 = 1 3  and  3 9 = 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaqGGa WaaSaaaeaacqGHsislcaaI3aaabaGaaGOmaiaaigdaaaGaeyypa0Za aSaaaeaacqGHsislcaaIXaaabaGaaG4maaaacaqGGaGaaeyyaiaab6 gacaqGKbGaaeiiaaqaaiaaykW7caaMc8UaaGPaVpaalaaabaGaaG4m aaqaaiaaiMdaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaG4maaaaaa aa@49F2@         [Converting into lowest term]

     1 3 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaae iiaiaaysW7caqGGaGaaeiiaiaabccadaWcaaqaaiabgkHiTiaaigda aeaacaaIZaaaaiabgcMi5oaalaaabaGaaGymaaqaaiaaiodaaaaaaa@40EE@

As     1 3 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaabc cacaqGGaWaaSaaaeaacqGHsislcaaIXaaabaGaaG4maaaacqGHGjsU daWcaaqaaiaaigdaaeaacaaIZaaaaaaa@3D85@  , therefore 7 21 3 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGOmaiaaigdaaaGaeyiyIK7aaSaaaeaacaaI ZaaabaGaaGyoaaaaaaa@3C64@  and does not represent same rational number.

 

 

(ii)          16 20  and  20 25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaGaaGOnaaqaaiaaikdacaaIWaaaaiaabccacaqGHbGa aeOBaiaabsgacaqGGaWaaSaaaeaacaaIYaGaaGimaaqaaiabgkHiTi aaikdacaaI1aaaaaaa@41B6@

16 20 = 4 5  and  20 25 = 4 5 = 4 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaigdacaaI2aaabaGaaGOmaiaaicdaaaGaeyypa0Za aSaaaeaacqGHsislcaaI0aaabaGaaGynaaaacaqGGaGaaeyyaiaab6 gacaqGKbGaaeiiaaqaamaalaaabaGaaGOmaiaaicdaaeaacqGHsisl caaIYaGaaGynaaaacqGH9aqpdaWcaaqaaiaaisdaaeaacqGHsislca aI1aaaaiabg2da9maalaaabaGaeyOeI0IaaGinaaqaaiaaiwdaaaaa aaa@4C3C@      [Converting into lowest term]

As 4 5 = 4 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabo hacaaMc8UaaGjbVpaalaaabaGaeyOeI0IaaGinaaqaaiaaiwdaaaGa eyypa0ZaaSaaaeaacqGHsislcaaI0aaabaGaaGynaaaaaaa@40A4@  therefore 16 20 = 20 25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaGaaGOnaaqaaiaaikdacaaIWaaaaiabg2da9maalaaa baGaaGOmaiaaicdaaeaacqGHsislcaaIYaGaaGynaaaaaaa@3EBA@  and represent same rational numbers.

 

(iii)      2 3  and  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaeyOeI0IaaG4maaaacaqGGaGaaeyyaiaab6ga caqGKbGaaeiiamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3EC6@

2 3 = 2 3  and 2 3 = 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaikdaaeaacqGHsislcaaIZaaaaiabg2da9maalaaa baGaaGOmaaqaaiaaiodaaaGaaeiiaiaabggacaqGUbGaaeizaaqaai aaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaaikdaaeaacaaIZaaa aiabg2da9maalaaabaGaaGOmaaqaaiaaiodaaaaaaaa@4973@                       [Converting into lowest term]

As 2 3 = 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabo hacaaMc8+aaSaaaeaacaaIYaaabaGaaG4maaaacqGH9aqpdaWcaaqa aiaaikdaaeaacaaIZaaaaaaa@3D35@  therefore   2 3 = 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaays W7daWcaaqaaiabgkHiTiaaikdaaeaacqGHsislcaaIZaaaaiabg2da 9maalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3DFA@ , and represent same rational number.

 

(iv)       3 5  and  12 20 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGymaiaaikdaaeaacaaIYaGaaGimaa aaaaa@403D@

3 5 = 3 5  and 12 20 = 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaaiodaaeaacaaI1aaaaiabg2da9maalaaabaGaeyOe I0IaaG4maaqaaiaaiwdaaaGaaeiiaiaabggacaqGUbGaaeizaaqaam aalaaabaGaeyOeI0IaaGymaiaaikdaaeaacaaIYaGaaGimaaaacqGH 9aqpdaWcaaqaaiabgkHiTiaaiodaaeaacaaI1aaaaaaaaa@469F@            [Converting into lowest term]

     3 5 = 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaae iiaiaabccacaqGGaGaaeiiaiaaysW7daWcaaqaaiabgkHiTiaaioda aeaacaaI1aaaaiabg2da9maalaaabaGaeyOeI0IaaG4maaqaaiaaiw daaaaaaa@4122@

     3 5 = 12 20 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaG4maaqaaiaa iwdaaaGaeyypa0ZaaSaaaeaacqGHsislcaaIXaGaaGOmaaqaaiaaik dacaaIWaaaaaaa@410B@

(v)          8 5  and  24 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aaabaGaeyOeI0IaaGynaaaacaqGGaGaaeyyaiaab6gacaqGKbGa aeiiamaalaaabaGaeyOeI0IaaGOmaiaaisdaaeaacaaIXaGaaGynaa aaaaa@4049@

8 5 = 8 5  and  24 15 = 8 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiIdaaeaacqGHsislcaaI1aaaaiabg2da9maalaaabaGaeyOe I0IaaGioaaqaaiaaiwdaaaGaaeiiaiaabggacaqGUbGaaeizaiaabc caaeaadaWcaaqaaiabgkHiTiaaikdacaaI0aaabaGaaGymaiaaiwda aaGaeyypa0ZaaSaaaeaacqGHsislcaaI4aaabaGaaGynaaaaaaaa@4757@           [Converting into lowest term]

     8 5 = 8 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaaG jbVlaabccacaqGGaGaaeiiaiaabccadaWcaaqaaiabgkHiTiaaiIda aeaacaaI1aaaaiabg2da9maalaaabaGaeyOeI0IaaGioaaqaaiaaiw daaaaaaa@412C@

     8 5 = 24 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiaiaaysW7daWcaaqaaiaaiIdaaeaacqGH sislcaaI1aaaaiabg2da9maalaaabaGaeyOeI0IaaGOmaiaaisdaae aacaaIXaGaaGynaaaaaaa@42A4@

(vi)       1 3  and  1 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaG4maaaacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiamaa laaabaGaeyOeI0IaaGymaaqaaiaaiMdaaaaaaa@3DDD@

1 3 = 1 3  and  1 9 = 1 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaMc8 +aaSaaaeaacaaIXaaabaGaaG4maaaacqGH9aqpdaWcaaqaaiaaigda aeaacaaIZaaaaiaabccacaqGHbGaaeOBaiaabsgacaqGGaaabaGaaG PaVpaalaaabaGaeyOeI0IaaGymaaqaaiaaiMdaaaGaeyypa0ZaaSaa aeaacqGHsislcaaIXaaabaGaaGyoaaaaaaaa@4709@               [Converting into lowest term]

     1 3 1 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaae iiaiaabccacaqGGaGaaeiiaiaaysW7daWcaaqaaiaaigdaaeaacaaI ZaaaaiabgcMi5oaalaaabaGaeyOeI0IaaGymaaqaaiaaiMdaaaaaaa@40F4@

     1 3 1 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiaiaaysW7daWcaaqaaiaaigdaaeaacaaI ZaaaaiabgcMi5oaalaaabaGaeyOeI0IaaGymaaqaaiaaiMdaaaaaaa@40F9@

(vii)    5 9  and  5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaeyOeI0IaaGyoaaaacaqGGaGaaeyyaiaab6ga caqGKbGaaeiiamaalaaabaGaaGynaaqaaiabgkHiTiaaiMdaaaaaaa@3FC5@

    5 9 = 5 9  and  5 9 = 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaI1aaabaGaeyOeI0Ia aGyoaaaacqGH9aqpdaWcaaqaaiaaiwdaaeaacaaI5aaaaiaabccaca qGHbGaaeOBaiaabsgacaqGGaWaaSaaaeaacaaI1aaabaGaeyOeI0Ia aGyoaaaacqGH9aqpdaWcaaqaaiaaiwdaaeaacqGHsislcaaI5aaaaa aa@4A28@               [Converting into lowest term]

      5 9 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSynIeLaae iiaiaabccacaqGGaGaaeiiaiaabccacaaMe8+aaSaaaeaacaaI1aaa baGaaGyoaaaacqGHGjsUdaWcaaqaaiaaiwdaaeaacqGHsislcaaI5a aaaaaa@41A5@

     5 9 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaae iiaiaabccacaqGGaGaaeiiaiaaysW7daWcaaqaaiabgkHiTiaaiwda aeaacqGHsislcaaI5aaaaiabgcMi5oaalaaabaGaaGynaaqaaiabgk HiTiaaiMdaaaaaaa@42E1@

Question: 7

Rewrite the following rational numbers in the simplest form:

(i)             8 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI4aaabaGaaGOnaaaaaaa@3857@

(ii)          25 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaaGynaaqaaiaaisdacaaI1aaaaaaa@38E0@  

(iii)      44 72 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aGaaGinaaqaaiaaiEdacaaIYaaaaaaa@39CE@  

(iv)       8 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI4aaabaGaaGymaiaaicdaaaaaaa@390C@

Solution

(i)             Given, 8 6 = 8÷2 6÷2 = 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaqGhb GaaeyAaiaabAhacaqGLbGaaeOBaiaabYcacaaMc8UaaGPaVpaalaaa baGaeyOeI0IaaGioaaqaaiaaiAdaaaaabaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8Uaeyypa0ZaaSaaaeaacqGHsislcaaI4aGaey49aGRaaGOmaa qaaiaaiAdacqGH3daUcaaIYaaaaaqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabg2da9maalaaabaGaeyOeI0IaaGinaaqaaiaaiodaaaaaaaa@72A4@            [H.C.F. of 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaaaa@369A@  and 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaaaa@3698@  is 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3694@  ]

(ii)          Given= 25 45 = 25÷5 45÷5 = 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaqGhb GaaeyAaiaabAhacaqGLbGaaeOBaiabg2da9maalaaabaGaaGOmaiaa iwdaaeaacaaI0aGaaGynaaaaaeaacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlabg2da9maalaaabaGaaGOmaiaaiwdacqGH3daUca aI1aaabaGaaGinaiaaiwdacqGH3daUcaaI1aaaaaqaaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0ZaaSaaaeaacaaI1aaa baGaaGyoaaaaaaaa@7643@           [H.C.F. of 25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiw daaaa@3753@  and 45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaiw daaaa@3755@  is 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3697@  ]

(iii)      Given, 44 72 = 44÷4 72÷4 = 11 18 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaqGhb GaaeyAaiaabAhacaqGLbGaaeOBaiaabYcadaWcaaqaaiabgkHiTiaa isdacaaI0aaabaGaaG4naiaaikdaaaaabaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlabg2da9maalaaabaGaeyOeI0IaaGinaiaaisdacq GH3daUcaaI0aaabaGaaG4naiaaikdacqGH3daUcaaI0aaaaaqaaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9aqpdaWcaaqaaiabgkHi TiaaigdacaaIXaaabaGaaGymaiaaiIdaaaaaaaa@770E@         [H.C.F. of 44 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaais daaaa@3754@  and 72 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaaik daaaa@3755@  is 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3696@  ]

(iv)       Given, 8 10 = 8÷2 10÷2 = 4 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaqGhb GaaeyAaiaabAhacaqGLbGaaeOBaiaabYcacaaMc8UaaGPaVpaalaaa baGaeyOeI0IaaGioaaqaaiaaigdacaaIWaaaaaqaaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0ZaaSaaaeaacqGHsislca aI4aGaey49aGRaaGOmaaqaaiaaigdacaaIWaGaey49aGRaaGOmaaaa aeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabg2da9maa laaabaGaeyOeI0IaaGinaaqaaiaaiwdaaaaaaaa@7A3C@           [H.C.F. of 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaaaa@369A@  and 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@374D@  is 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3694@  ]

Question: 8

Fill in the boxes with the correct symbol out of >,<, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOpa4Jaai ilaiaaysW7cqGH8aapcaGGSaaaaa@3AD1@  and = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DE@ .

(i)             5 7        2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaG4naaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiaaikdaaeaacaaIZaaaaa aa@3DF7@

(ii)          4 5        5 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiabgkHiTiaaiwdaaeaaca aI3aaaaaaa@3EE8@

(iii)      7 8        14 16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGioaaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiaaigdacaaI0aaabaGaey OeI0IaaGymaiaaiAdaaaaaaa@4062@

(iv)       8 5        7 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI4aaabaGaaGynaaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiabgkHiTiaaiEdaaeaaca aI0aaaaaaa@3EEB@

(v)          1 3        1 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaeyOeI0IaaG4maaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiabgkHiTiaaigdaaeaaca aI0aaaaaaa@3EDC@

(vi)       5 11        5 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaeyOeI0IaaGymaiaaigdaaaWaauIhaeaacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccaaaWaaSaaaeaacqGHsislcaaI1a aabaGaaGymaiaaigdaaaaaaa@4055@

(vii)    0        7 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimamaaL4 babaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaaaamaalaaa baGaeyOeI0IaaG4naaqaaiaaiAdaaaaaaa@3D29@

Solution

(i)             5 7  <  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaG4naaaadaqjEaqaaiaabccacqGH8aapcaqG GaaaamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3C6F@        

Since, the positive number is always greater than the negative number.

Hence, 5 7  <  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaG4naaaadaqjEaqaaiaabccacqGH8aapcaqG GaaaamaalaaabaGaaGOmaaqaaiaaiodaaaaaaa@3C6F@

(ii)          4 5        5 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiabgkHiTiaaiwdaaeaaca aI3aaaaaaa@3EE8@

Making denominator same for both the rational number.

4×7 5×7        5×5 7×5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aGaey41aqRaaG4naaqaaiaaiwdacqGHxdaTcaaI3aaa amaaL4babaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaaaam aalaaabaGaeyOeI0IaaGynaiabgEna0kaaiwdaaeaacaaI3aGaey41 aqRaaGynaaaaaaa@4A44@

             28 35  <  25 35 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaGOmaiaaiIda aeaacaaIZaGaaGynaaaadaqjEaqaaiaabccacqGH8aapcaqGGaaaam aalaaabaGaeyOeI0IaaGOmaiaaiwdaaeaacaaIZaGaaGynaaaaaaa@453D@

        As, -25 > than -28

Hence,  4 5  <  5 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeisaiaabw gacaqGUbGaae4yaiaabwgacaqGSaGaaeiiamaalaaabaGaeyOeI0Ia aGinaaqaaiaaiwdaaaWaauIhaeaacaqGGaGaeyipaWJaaeiiaaaada WcaaqaaiabgkHiTiaaiwdaaeaacaaI3aaaaaaa@4324@

(iii)      7 8        14 16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGioaaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiaaigdacaaI0aaabaGaey OeI0IaaGymaiaaiAdaaaaaaa@4062@

Making denominator same for both the rational number.

7×2 8×2        14×(1) 16×(1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aGaey41aqRaaGOmaaqaaiaaiIdacqGHxdaTcaaIYaaa amaaL4babaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaaaam aalaaabaGaaGymaiaaisdacqGHxdaTcaGGOaGaeyOeI0IaaGymaiaa cMcaaeaacqGHsislcaaIXaGaaGOnaiabgEna0kaacIcacqGHsislca aIXaGaaiykaaaaaaa@5038@

     14 16  =  14 16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaGymaiaaisda aeaacaaIXaGaaGOnaaaadaqjEaqaaiaabccacqGH9aqpcaqGGaaaam aalaaabaGaeyOeI0IaaGymaiaaisdaaeaacaaIXaGaaGOnaaaaaaa@4536@

     7 8  =  14 16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaG4naaqaaiaa iIdaaaWaauIhaeaacaqGGaGaeyypa0JaaeiiaaaadaWcaaqaaiaaig dacaaI0aaabaGaeyOeI0IaaGymaiaaiAdaaaaaaa@43C5@

(iv)       8 5        7 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI4aaabaGaaGynaaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiabgkHiTiaaiEdaaeaaca aI0aaaaaaa@3EEB@

Making denominator same for both the rational number.

8×4 5×4        7×5 4×5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI4aGaey41aqRaaGinaaqaaiaaiwdacqGHxdaTcaaI0aaa amaaL4babaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaaaam aalaaabaGaeyOeI0IaaG4naiabgEna0kaaiwdaaeaacaaI0aGaey41 aqRaaGynaaaaaaa@4A41@

     32 20  >  35 20 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaG4maiaaikda aeaacaaIYaGaaGimaaaadaqjEaqaaiaabccacqGH+aGpcaqGGaaaam aalaaabaGaeyOeI0IaaG4maiaaiwdaaeaacaaIYaGaaGimaaaaaaa@4531@

     8 5  >  7 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaaeiiamaalaaabaGaeyOeI0IaaGioaaqaaiaa iwdaaaWaauIhaeaacaqGGaGaeyOpa4JaaeiiaaaadaWcaaqaaiabgk HiTiaaiEdaaeaacaaI0aaaaaaa@4250@

(v)          1 3        1 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaeyOeI0IaaG4maaaadaqjEaqaaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaaaadaWcaaqaaiabgkHiTiaaigdaaeaaca aI0aaaaaaa@3EDC@

        Making denominator same for both the rational numbers.

        1×4 3×4        1×3 4×3 4 12  <   3 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaigdacqGHxdaTcaaI0aaabaGaeyOeI0IaaG4maiabgEna0kaa isdaaaWaauIhaeaacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc caaaWaaSaaaeaacqGHsislcaaIXaGaey41aqRaaG4maaqaaiaaisda cqGHxdaTcaaIZaaaaaqaaiabgkHiTmaalaaabaGaaGinaaqaaiaaig dacaaIYaaaamaaL4babaGaaeiiaiabgYda8iaabccacaqGGaaaamaa laaabaGaeyOeI0IaaG4maaqaaiaaigdacaaIYaaaaaaaaa@53CB@

          1 3  <  1 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiamaalaaabaGaeyOeI0IaaGymaaqaaiaaiodaaaWaauIhaeaacaqG GaGaeyipaWJaaeiiaaaadaWcaaqaaiabgkHiTiaaigdaaeaacaaI0a aaaaaa@4054@

(vi)       5 11        5 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaeyOeI0IaaGymaiaaigdaaaWaauIhaeaacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccaaaWaaSaaaeaacqGHsislcaaI1a aabaGaaGymaiaaigdaaaaaaa@4055@

     5 11  =  5 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaGaaeiiamaalaaabaGaaGynaaqaaiabgkHiTiaa igdacaaIXaaaamaaL4babaGaaeiiaiabg2da9iaabccaaaWaaSaaae aacqGHsislcaaI1aaabaGaaGymaiaaigdaaaaaaa@43B8@

(vii)    0  >  7 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimamaaL4 babaGaaeiiaiabg6da+iaabccaaaWaaSaaaeaacqGHsislcaaI3aaa baGaaGOnaaaaaaa@3BA5@         

Since, 0 is greater than every negative number.

Question: 9

Which is greater in each of the following:

(i)             2 3 , 5 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaaG4maaaacaGGSaWaaSaaaeaacaaI1aaabaGaaGOmaaaa aaa@399C@

(ii)          5 6 , 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGOnaaaacaGGSaWaaSaaaeaacqGHsislcaaI 0aaabaGaaG4maaaaaaa@3B7C@

(iii)      3 4 , 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGinaaaacaGGSaWaaSaaaeaacaaIYaaabaGa eyOeI0IaaG4maaaaaaa@3B76@

(iv)       1 4 , 1 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGinaaaacaGGSaWaaSaaaeaacaaIXaaabaGa aGinaaaaaaa@3A87@

(v)          3 2 7 ,3 4 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4mamaalaaabaGaaGOmaaqaaiaaiEdaaaGaaiilaiabgkHiTiaaioda daWcaaqaaiaaisdaaeaacaaI1aaaaaaa@3CF6@

Solution

(i)             2 3 , 5 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaaG4maaaacaGGSaWaaSaaaeaacaaI1aaabaGaaGOmaaaa aaa@399C@

Making denominator same for the rational numbers.

2×2 3×2 = 4 6  and  5×3 2×3 = 15 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaey41aqRaaGOmaaqaaiaaiodacqGHxdaTcaaIYaaaaiabg2da 9maalaaabaGaaGinaaqaaiaaiAdaaaGaaeiiaiaabggacaqGUbGaae izaiaabccadaWcaaqaaiaaiwdacqGHxdaTcaaIZaaabaGaaGOmaiab gEna0kaaiodaaaGaeyypa0ZaaSaaaeaacaaIXaGaaGynaaqaaiaaiA daaaaaaa@4E20@

Since 4 6  <  15 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aaabaGaaGOnaaaadaqjEaqaaiaabccacqGH8aapcaqGGaaaamaa laaabaGaaGymaiaaiwdaaeaacaaI2aaaaaaa@3C41@      

Therefore 2 3  <  5 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaaG4maaaadaqjEaqaaiaabccacqGH8aapcaqGGaaaamaa laaabaGaaGynaaqaaiaaikdaaaaaaa@3B7D@

(ii)          5 6 , 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGOnaaaacaGGSaWaaSaaaeaacqGHsislcaaI 0aaabaGaaG4maaaaaaa@3B7C@

Making denominator same for both the rational numbers.

5×1 6×1 = 5 6  and  4×2 3×2 = 8 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aGaey41aqRaaGymaaqaaiaaiAdacqGHxdaTcaaIXaaa aiabg2da9maalaaabaGaeyOeI0IaaGynaaqaaiaaiAdaaaGaaeiiai aabggacaqGUbGaaeizaiaabccadaWcaaqaaiabgkHiTiaaisdacqGH xdaTcaaIYaaabaGaaG4maiabgEna0kaaikdaaaGaeyypa0ZaaSaaae aacqGHsislcaaI4aaabaGaaGOnaaaaaaa@511F@

Since 5 6  >  8 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGOnaaaadaqjEaqaaiaabccacqGH+aGpcaqG GaaaamaalaaabaGaeyOeI0IaaGioaaqaaiaaiAdaaaaaaa@3D68@    

Therefore 5 6  >  4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI1aaabaGaaGOnaaaadaqjEaqaaiaabccacqGH+aGpcaqG GaaaamaalaaabaGaeyOeI0IaaGinaaqaaiaaiodaaaaaaa@3D61@

(iii)      3 4 , 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGinaaaacaGGSaWaaSaaaeaacaaIYaaabaGa eyOeI0IaaG4maaaaaaa@3B76@

Making denominator same for both the rational numbers.

3×3 4×3 = 9 12  and  2×(4) 3×(4) = 8 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaGaey41aqRaaG4maaqaaiaaisdacqGHxdaTcaaIZaaa aiabg2da9maalaaabaGaeyOeI0IaaGyoaaqaaiaaigdacaaIYaaaai aabccacaqGHbGaaeOBaiaabsgacaqGGaWaaSaaaeaacaaIYaGaey41 aqRaaiikaiabgkHiTiaaisdacaGGPaaabaGaeyOeI0IaaG4maiabgE na0kaacIcacqGHsislcaaI0aGaaiykaaaacqGH9aqpdaWcaaqaaiab gkHiTiaaiIdaaeaacaaIXaGaaGOmaaaaaaa@571F@

Since 9 12  <  8 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI5aaabaGaaGymaiaaikdaaaWaauIhaeaacaqGGaGaeyip aWJaaeiiaaaadaWcaaqaaiabgkHiTiaaiIdaaeaacaaIXaGaaGOmaa aaaaa@3ED6@    

Therefore 3 4  <  2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGinaaaadaqjEaqaaiaabccacqGH8aapcaqG GaaaamaalaaabaGaaGOmaaqaaiabgkHiTiaaiodaaaaaaa@3D57@

(iv)       1 4  <  1 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGinaaaadaqjEaqaaiaabccacqGH8aapcaqG GaaaamaalaaabaGaaGymaaqaaiaaisdaaaaaaa@3C68@  Since positive number is always greater than negative number.

(v)          3 2 7 ,3 4 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4mamaalaaabaGaaGOmaaqaaiaaiEdaaaGaaiilaiabgkHiTiaaioda daWcaaqaaiaaisdaaeaacaaI1aaaaaaa@3CF6@

3 2 7 = 23 7 = 23×5 7×5 = 115 35  and 3 4 5 = 19 5 = 19×7 5×7 = 133 35 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGHsi slcaaIZaWaaSaaaeaacaaIYaaabaGaaG4naaaacqGH9aqpdaWcaaqa aiabgkHiTiaaikdacaaIZaaabaGaaG4naaaacqGH9aqpdaWcaaqaai abgkHiTiaaikdacaaIZaGaey41aqRaaGynaaqaaiaaiEdacqGHxdaT caaI1aaaaiabg2da9maalaaabaGaeyOeI0IaaGymaiaaigdacaaI1a aabaGaaG4maiaaiwdaaaGaaeiiaiaabggacaqGUbGaaeizaiaaysW7 aeaacqGHsislcaaIZaWaaSaaaeaacaaI0aaabaGaaGynaaaacqGH9a qpdaWcaaqaaiabgkHiTiaaigdacaaI5aaabaGaaGynaaaacqGH9aqp daWcaaqaaiabgkHiTiaaigdacaaI5aGaey41aqRaaG4naaqaaiaaiw dacqGHxdaTcaaI3aaaaiabg2da9maalaaabaGaeyOeI0IaaGymaiaa iodacaaIZaaabaGaaG4maiaaiwdaaaaaaaa@68F6@

Since 115 35  >  133 35 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaGaaGymaiaaiwdaaeaacaaIZaGaaGynaaaadaqjEaqa aiaabccacqGH+aGpcaqGGaaaamaalaaabaGaeyOeI0IaaGymaiaaio dacaaIZaaabaGaaG4maiaaiwdaaaaaaa@41C9@

Therefore 3 2 7  >  3 4 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4mamaalaaabaGaaGOmaaqaaiaaiEdaaaWaauIhaeaacaqGGaGaeyOp a4JaaeiiaaaacqGHsislcaaIZaWaaSaaaeaacaaI0aaabaGaaGynaa aaaaa@3EDB@

Question: 10

Write the following rational numbers in ascending order:

(i)             3 5 , 2 5 , 1 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaIXaaabaGaaG ynaaaaaaa@3EA0@

(ii)          1 3 , 2 9 , 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaG4maaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaGyoaaaacaGGSaWaaSaaaeaacqGHsislcaaI0aaabaGaaG 4maaaaaaa@3EA1@

(iii)      3 7 , 3 2 , 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaG4naaaacaGGSaWaaSaaaeaacqGHsislcaaI ZaaabaGaaGOmaaaacaGGSaWaaSaaaeaacqGHsislcaaIZaaabaGaaG inaaaaaaa@3EA1@

Solution

(i)             3 5 , 2 5 , 1 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaGynaaaacaGGSaWaaSaaaeaacqGHsislcaaIXaaabaGaaG ynaaaaaaa@3EA0@

As,3<2<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabo hacaqGSaGaeyOeI0IaaG4maiabgYda8iabgkHiTiaaikdacqGH8aap cqGHsislcaaIXaaaaa@3F45@  

Hence,  3 5 < 2 5 < 1 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeisaiaabw gacaqGUbGaae4yaiaabwgacaqGSaGaaeiiamaalaaabaGaeyOeI0Ia aG4maaqaaiaaiwdaaaGaeyipaWZaaSaaaeaacqGHsislcaaIYaaaba GaaGynaaaacqGH8aapdaWcaaqaaiabgkHiTiaaigdaaeaacaaI1aaa aaaa@450C@

(ii)          1 3 , 2 9 , 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaG4maaaacaGGSaWaaSaaaeaacqGHsislcaaI YaaabaGaaGyoaaaacaGGSaWaaSaaaeaacqGHsislcaaI0aaabaGaaG 4maaaaaaa@3EA1@

1×3 3×3 , 2 9 , 4×3 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaGaey41aqRaaG4maaqaaiaaiodacqGHxdaTcaaIZaaa aiaacYcadaWcaaqaaiabgkHiTiaaikdaaeaacaaI5aaaaiaacYcada WcaaqaaiabgkHiTiaaisdacqGHxdaTcaaIZaaabaGaaG4maiabgEna 0kaaiodaaaaaaa@49F1@  [Converting these into like fractions]

    3 9 , 2 9 , 12 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIZaaabaGaaGyoaaaa caGGSaWaaSaaaeaacqGHsislcaaIYaaabaGaaGyoaaaacaGGSaWaaS aaaeaacqGHsislcaaIXaGaaGOmaaqaaiaaiMdaaaaaaa@43AE@    

As,12<3<2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabo hacaqGSaGaaGPaVlabgkHiTiaaigdacaaIYaGaeyipaWJaeyOeI0Ia aG4maiabgYda8iabgkHiTiaaikdaaaa@418C@  

4 3 < 1 3 < 2 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIW1aaS aaaeaacqGHsislcaaI0aaabaGaaG4maaaacqGH8aapdaWcaaqaaiab gkHiTiaaigdaaeaacaaIZaaaaiabgYda8maalaaabaGaeyOeI0IaaG OmaaqaaiaaiMdaaaaaaa@4088@  

(iii)      3 7 , 3 2 , 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaG4naaaacaGGSaWaaSaaaeaacqGHsislcaaI ZaaabaGaaGOmaaaacaGGSaWaaSaaaeaacqGHsislcaaIZaaabaGaaG inaaaaaaa@3EA1@

[Converting these into like fractions]

3×4 7×4 , 3×14 2×14 , 3×7 4×7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaGaey41aqRaaGinaaqaaiaaiEdacqGHxdaTcaaI0aaa aiaacYcadaWcaaqaaiabgkHiTiaaiodacqGHxdaTcaaIXaGaaGinaa qaaiaaikdacqGHxdaTcaaIXaGaaGinaaaacaGGSaWaaSaaaeaacqGH sislcaaIZaGaey41aqRaaG4naaqaaiaaisdacqGHxdaTcaaI3aaaaa aa@511B@

12 28 , 42 28 , 21 28 As,-42<21<12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaigdacaaIYaaabaGaaGOmaiaaiIdaaaGaaiilamaa laaabaGaeyOeI0IaaGinaiaaikdaaeaacaaIYaGaaGioaaaacaGGSa WaaSaaaeaacqGHsislcaaIYaGaaGymaaqaaiaaikdacaaI4aaaaaqa aiaabgeacaqGZbGaaeilaiaaykW7caqGTaGaaeinaiaabkdacqGH8a apcqGHsislcaaIYaGaaGymaiabgYda8iabgkHiTiaaigdacaaIYaaa aaa@4FF7@

    3 2 < 3 4 < 3 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae iiaiaabccacaqGGaWaaSaaaeaacqGHsislcaaIZaaabaGaaGOmaaaa cqGH8aapdaWcaaqaaiabgkHiTiaaiodaaeaacaaI0aaaaiabgYda8m aalaaabaGaeyOeI0IaaG4maaqaaiaaiEdaaaaaaa@438F@

 

Exercise: 9.2 (4)

 

Question: 1

Find the sum:

(i)             5 4 +( 11 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaGinaaaacqGHRaWkdaqadaqaamaalaaabaGaeyOeI0Ia aGymaiaaigdaaeaacaaI0aaaaaGaayjkaiaawMcaaaaa@3D01@

(ii)          5 3 + 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaG4maaaacqGHRaWkdaWcaaqaaiaaiodaaeaacaaI1aaa aaaa@39D2@

(iii)      9 10 + 22 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI5aaabaGaaGymaiaaicdaaaGaey4kaSYaaSaaaeaacaaI YaGaaGOmaaqaaiaaigdacaaI1aaaaaaa@3CF1@

(iv)       3 11 + 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaeyOeI0IaaGymaiaaigdaaaGaey4kaSYaaSaa aeaacaaI1aaabaGaaGyoaaaaaaa@3C69@

(v)          8 19 + (2) 57 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI4aaabaGaaGymaiaaiMdaaaGaey4kaSYaaSaaaeaacaGG OaGaeyOeI0IaaGOmaiaacMcaaeaacaaI1aGaaG4naaaaaaa@3E89@

(vi)       2 3 +0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4maaaacqGHRaWkcaaIWaaaaa@39EA@

(vii)    2 1 3 +4 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmamaalaaabaGaaGymaaqaaiaaiodaaaGaey4kaSIaaGinamaalaaa baGaaG4maaqaaiaaiwdaaaaaaa@3C35@

Solution

(i)  

5 4 +( 11 4 ) = 511 4 = 6 4 = 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaadaWcaa qaaiaaiwdaaeaacaaI0aaaaiabgUcaRmaabmaabaWaaSaaaeaacqGH sislcaaIXaGaaGymaaqaaiaaisdaaaaacaGLOaGaayzkaaaabaGaey ypa0ZaaSaaaeaacaaI1aGaeyOeI0IaaGymaiaaigdaaeaacaaI0aaa aaqaaiabg2da9maalaaabaGaeyOeI0IaaGOnaaqaaiaaisdaaaaaba Gaeyypa0ZaaSaaaeaacqGHsislcaaIZaaabaGaaGOmaaaaaaaa@48FF@

(ii)           

 

5 3 + 3 5 = 5×5 3×5 + 3×3 5×3 = 25 15 + 9 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaadaWcaa qaaiaaiwdaaeaacaaIZaaaaiabgUcaRmaalaaabaGaaG4maaqaaiaa iwdaaaGaeyypa0ZaaSaaaeaacaaI1aGaey41aqRaaGynaaqaaiaaio dacqGHxdaTcaaI1aaaaiabgUcaRmaalaaabaGaaG4maiabgEna0kaa iodaaeaacaaI1aGaey41aqRaaG4maaaaaeaacqGH9aqpdaWcaaqaai aaikdacaaI1aaabaGaaGymaiaaiwdaaaGaey4kaSYaaSaaaeaacaaI 5aaabaGaaGymaiaaiwdaaaaaaaa@5169@              [L.C.M. of 3 and 5 is 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaaGynaiaabccacaqGPbGaae4C aiaabccacaaIXaGaaGynaaaa@3FF8@  ]

= 25+9 15 = 34 15 =2 4 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaacqGH9a qpdaWcaaqaaiaaikdacaaI1aGaey4kaSIaaGyoaaqaaiaaigdacaaI 1aaaaaqaaiabg2da9maalaaabaGaaG4maiaaisdaaeaacaaIXaGaaG ynaaaaaeaacqGH9aqpcaaIYaWaaSaaaeaacaaI0aaabaGaaGymaiaa iwdaaaaaaaa@43A7@

(iii)       

9 10 + 22 15 = 9×3 10×3 + 22×2 15×2 = 27 30 + 44 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaadaWcaa qaaiabgkHiTiaaiMdaaeaacaaIXaGaaGimaaaacqGHRaWkdaWcaaqa aiaaikdacaaIYaaabaGaaGymaiaaiwdaaaaabaGaeyypa0ZaaSaaae aacqGHsislcaaI5aGaey41aqRaaG4maaqaaiaaigdacaaIWaGaey41 aqRaaG4maaaacqGHRaWkdaWcaaqaaiaaikdacaaIYaGaey41aqRaaG OmaaqaaiaaigdacaaI1aGaey41aqRaaGOmaaaaaeaacqGH9aqpdaWc aaqaaiabgkHiTiaaikdacaaI3aaabaGaaG4maiaaicdaaaGaey4kaS YaaSaaaeaacaaI0aGaaGinaaqaaiaaiodacaaIWaaaaaaaaa@5944@   [L.C.M. of 10 and 15 is 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaaigdacaaI1aGaaeii aiaabMgacaqGZbGaaeiiaiaaiodacaaIWaaaaa@4168@  ]

        = 27+44 30 = 17 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaikdacaaI3aGaey4kaSIaaGinaiaaisda aeaacaaIZaGaaGimaaaaaeaacqGH9aqpdaWcaaqaaiaaigdacaaI3a aabaGaaG4maiaaicdaaaaaaaa@413D@

(iv)        

    3 11 + 5 9 = 3×9 11×9 + 5×11 9×11 = 27 99 + 55 99 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabaeaadaWcaa qaaiabgkHiTiaaiodaaeaacqGHsislcaaIXaGaaGymaaaacqGHRaWk daWcaaqaaiaaiwdaaeaacaaI5aaaaaqaaiabg2da9maalaaabaGaey OeI0IaaG4maiabgEna0kaaiMdaaeaacqGHsislcaaIXaGaaGymaiab gEna0kaaiMdaaaGaey4kaSYaaSaaaeaacaaI1aGaey41aqRaaGymai aaigdaaeaacaaI5aGaey41aqRaaGymaiaaigdaaaaabaGaeyypa0Za aSaaaeaacaaIYaGaaG4naaqaaiaaiMdacaaI5aaaaiabgUcaRmaala aabaGaaGynaiaaiwdaaeaacaaI5aGaaGyoaaaaaaaa@58E7@     [L.C.M. of 11 and 9 is 99 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaig dacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaaiMdacaqGGaGaaeyA aiaabohacaqGGaGaaGyoaiaaiMdaaaa@40C1@  ]

        = 27+55 99 = 82 99 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaikdacaaI3aGaey4kaSIaaGynaiaaiwdaaeaacaaI 5aGaaGyoaaaaaeaacqGH9aqpdaWcaaqaaiaaiIdacaaIYaaabaGaaG yoaiaaiMdaaaaaaaa@4072@

(v)           

    8 19 + (2) 57 = 8×3 19×3 + (2)×1 57×1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaiIdaaeaacaaIXaGaaGyoaaaacqGHRaWkdaWcaaqa aiaacIcacqGHsislcaaIYaGaaiykaaqaaiaaiwdacaaI3aaaaaqaai abg2da9maalaaabaGaeyOeI0IaaGioaiabgEna0kaaiodaaeaacaaI XaGaaGyoaiabgEna0kaaiodaaaGaey4kaSYaaSaaaeaacaGGOaGaey OeI0IaaGOmaiaacMcacqGHxdaTcaaIXaaabaGaaGynaiaaiEdacqGH xdaTcaaIXaaaaaaaaa@5393@    [L.C.M. of 19 and 57 is 57 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiM dacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaaiwdacaaI3aGaaeii aiaabMgacaqGZbGaaeiiaiaaiwdacaaI3aaaaa@4180@  ]

        = 24 57 + (2) 57 = 242 57 = 26 57 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqGH9a qpdaWcaaqaaiabgkHiTiaaikdacaaI0aaabaGaaGynaiaaiEdaaaGa ey4kaSYaaSaaaeaacaGGOaGaeyOeI0IaaGOmaiaacMcaaeaacaaI1a GaaG4naaaaaeaacqGH9aqpdaWcaaqaaiabgkHiTiaaikdacaaI0aGa eyOeI0IaaGOmaaqaaiaaiwdacaaI3aaaaaqaaiabg2da9maalaaaba GaeyOeI0IaaGOmaiaaiAdaaeaacaaI1aGaaG4naaaaaaaa@4BFA@

(vi)        

2 3 +0= 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaG4maaaacqGHRaWkcaaIWaGaeyypa0ZaaSaa aeaacqGHsislcaaIYaaabaGaaG4maaaaaaa@3D66@

 

(vii)     

2 1 3 +4 3 5 = 7 3 + 23 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGHsi slcaaIYaWaaSaaaeaacaaIXaaabaGaaG4maaaacqGHRaWkcaaI0aWa aSaaaeaacaaIZaaabaGaaGynaaaaaeaacqGH9aqpdaWcaaqaaiabgk HiTiaaiEdaaeaacaaIZaaaaiabgUcaRmaalaaabaGaaGOmaiaaioda aeaacaaI1aaaaaaaaa@42E6@             [L.C.M. of 3 and 5 is 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaaGynaiaabccacaqGPbGaae4C aiaabccacaaIXaGaaGynaaaa@3FF8@  ]

        = 7×5 3×5 + 23×3 5×3 = 35 15 + 69 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaiEdacqGHxdaTcaaI1aaabaGaaG4maiab gEna0kaaiwdaaaGaey4kaSYaaSaaaeaacaaIYaGaaG4maiabgEna0k aaiodaaeaacaaI1aGaey41aqRaaG4maaaaaeaacqGH9aqpdaWcaaqa aiabgkHiTiaaiodacaaI1aaabaGaaGymaiaaiwdaaaGaey4kaSYaaS aaaeaacaaI2aGaaGyoaaqaaiaaigdacaaI1aaaaaaaaa@50C6@

 

         = 35+69 15 = 34 15 =2 4 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaiodacaaI1aGaey4kaSIaaGOnaiaaiMda aeaacaaIXaGaaGynaaaaaeaacqGH9aqpdaWcaaqaaiaaiodacaaI0a aabaGaaGymaiaaiwdaaaaabaGaeyypa0JaaGOmamaalaaabaGaaGin aaqaaiaaigdacaaI1aaaaaaaaa@4553@

Question: 2

Find

(i)             7 24 17 36 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aaabaGaaGOmaiaaisdaaaGaeyOeI0YaaSaaaeaacaaIXaGaaG4n aaqaaiaaiodacaaI2aaaaaaa@3C19@

(ii)          5 63 ( 6 21 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaGOnaiaaiodaaaGaeyOeI0YaaeWaaeaadaWcaaqaaiab gkHiTiaaiAdaaeaacaaIYaGaaGymaaaaaiaawIcacaGLPaaaaaa@3DCE@

(iii)      6 13 ( 7 15 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI2aaabaGaaGymaiaaiodaaaGaeyOeI0YaaeWaaeaadaWc aaqaaiabgkHiTiaaiEdaaeaacaaIXaGaaGynaaaaaiaawIcacaGLPa aaaaa@3EBB@

(iv)       3 8 7 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGioaaaacqGHsisldaWcaaqaaiaaiEdaaeaa caaIXaGaaGymaaaaaaa@3B88@

(v)          2 1 9 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmamaalaaabaGaaGymaaqaaiaaiMdaaaGaeyOeI0IaaGOnaaaa@3ABC@

Solution

(i)             7 24 17 36 = 7×3 24×3 17×2 36×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiEdaaeaacaaIYaGaaGinaaaacqGHsisldaWcaaqaaiaaigda caaI3aaabaGaaG4maiaaiAdaaaaabaGaeyypa0ZaaSaaaeaacaaI3a Gaey41aqRaaG4maaqaaiaaikdacaaI0aGaey41aqRaaG4maaaacqGH sisldaWcaaqaaiaaigdacaaI3aGaey41aqRaaGOmaaqaaiaaiodaca aI2aGaey41aqRaaGOmaaaaaaaa@4EB4@              

[L.C.M. of 24 and 36 is 72 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaais dacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaaiodacaaI2aGaaeii aiaabMgacaqGZbGaaeiiaiaaiEdacaaIYaaaaa@4176@  ]

        = 21 72 34 72 = 2134 72 = 13 72 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacqGH9a qpdaWcaaqaaiaaikdacaaIXaaabaGaaG4naiaaikdaaaGaeyOeI0Ya aSaaaeaacaaIZaGaaGinaaqaaiaaiEdacaaIYaaaaaqaaiabg2da9m aalaaabaGaaGOmaiaaigdacqGHsislcaaIZaGaaGinaaqaaiaaiEda caaIYaaaaaqaaiabg2da9maalaaabaGaeyOeI0IaaGymaiaaiodaae aacaaI3aGaaGOmaaaaaaaa@494C@

(ii)          5 63 ( 6 21 ) = 5×1 63×1 ( 6×3 21×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiwdaaeaacaaI2aGaaG4maaaacqGHsisldaqadaqaamaalaaa baGaeyOeI0IaaGOnaaqaaiaaikdacaaIXaaaaaGaayjkaiaawMcaaa qaaiabg2da9maalaaabaGaaGynaiabgEna0kaaigdaaeaacaaI2aGa aG4maiabgEna0kaaigdaaaGaeyOeI0YaaeWaaeaadaWcaaqaaiabgk HiTiaaiAdacqGHxdaTcaaIZaaabaGaaGOmaiaaigdacqGHxdaTcaaI ZaaaaaGaayjkaiaawMcaaaaaaa@521C@           

[L.C.M. of 63 and 21 is 63 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaio dacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaaikdacaaIXaGaaeii aiaabMgacaqGZbGaaeiiaiaaiAdacaaIZaaaaa@4173@  ]

        = 5 63 18 63 = 5(18) 63 = 5+18 63 = 23 63 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaiwdaaeaacaaI2aGaaG4maaaacqGHsisldaWcaaqa aiabgkHiTiaaigdacaaI4aaabaGaaGOnaiaaiodaaaaabaGaeyypa0 ZaaSaaaeaacaaI1aGaeyOeI0IaaiikaiabgkHiTiaaigdacaaI4aGa aiykaaqaaiaaiAdacaaIZaaaaaqaaiabg2da9maalaaabaGaaGynai abgUcaRiaaigdacaaI4aaabaGaaGOnaiaaiodaaaaabaGaeyypa0Za aSaaaeaacaaIYaGaaG4maaqaaiaaiAdacaaIZaaaaaaaaa@4FD6@

(iii)      6 13 ( 7 15 ) = 6×15 13×15 ( 7×13 15×13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaiAdaaeaacaaIXaGaaG4maaaacqGHsisldaqadaqa amaalaaabaGaeyOeI0IaaG4naaqaaiaaigdacaaI1aaaaaGaayjkai aawMcaaaqaaiabg2da9maalaaabaGaeyOeI0IaaGOnaiabgEna0kaa igdacaaI1aaabaGaaGymaiaaiodacqGHxdaTcaaIXaGaaGynaaaacq GHsisldaqadaqaamaalaaabaGaeyOeI0IaaG4naiabgEna0kaaigda caaIZaaabaGaaGymaiaaiwdacqGHxdaTcaaIXaGaaG4maaaaaiaawI cacaGLPaaaaaaa@56EA@      

[L.C.M. of 13 and 15 is 195 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaio dacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaaigdacaaI1aGaaeii aiaabMgacaqGZbGaaeiiaiaaigdacaaI5aGaaGynaaaa@4231@  ]

        = 90 195 ( 91 195 ) = 90(91) 195 = 90+91 195 = 1 195 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaiMdacaaIWaaabaGaaGymaiaaiMdacaaI 1aaaaiabgkHiTmaabmaabaWaaSaaaeaacqGHsislcaaI5aGaaGymaa qaaiaaigdacaaI5aGaaGynaaaaaiaawIcacaGLPaaaaeaacqGH9aqp daWcaaqaaiabgkHiTiaaiMdacaaIWaGaeyOeI0IaaiikaiabgkHiTi aaiMdacaaIXaGaaiykaaqaaiaaigdacaaI5aGaaGynaaaaaeaacqGH 9aqpdaWcaaqaaiabgkHiTiaaiMdacaaIWaGaey4kaSIaaGyoaiaaig daaeaacaaIXaGaaGyoaiaaiwdaaaaabaGaeyypa0ZaaSaaaeaacaaI XaaabaGaaGymaiaaiMdacaaI1aaaaaaaaa@5965@

(iv)       3 8 7 11 = 3×11 8×11 7×8 11×8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaiodaaeaacaaI4aaaaiabgkHiTmaalaaabaGaaG4n aaqaaiaaigdacaaIXaaaaaqaaiabg2da9maalaaabaGaeyOeI0IaaG 4maiabgEna0kaaigdacaaIXaaabaGaaGioaiabgEna0kaaigdacaaI XaaaaiabgkHiTmaalaaabaGaaG4naiabgEna0kaaiIdaaeaacaaIXa GaaGymaiabgEna0kaaiIdaaaaaaaa@4F10@            

[L.C.M. of 8 and 11 is 88 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaaGymaiaaigdacaqGGaGaaeyA aiaabohacaqGGaGaaGioaiaaiIdaaaa@40BE@  ]

= 33 88 56 88 = 3356 88 = 89 88 =1 1 88 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaiodacaaIZaaabaGaaGioaiaaiIdaaaGa eyOeI0YaaSaaaeaacaaI1aGaaGOnaaqaaiaaiIdacaaI4aaaaaqaai abg2da9maalaaabaGaeyOeI0IaaG4maiaaiodacqGHsislcaaI1aGa aGOnaaqaaiaaiIdacaaI4aaaaaqaaiabg2da9maalaaabaGaeyOeI0 IaaGioaiaaiMdaaeaacaaI4aGaaGioaaaaaeaacqGH9aqpcqGHsisl caaIXaWaaSaaaeaacaaIXaaabaGaaGioaiaaiIdaaaaaaaa@5058@

 

(v)          2 1 9 6= 19 9 6 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmamaalaaabaGaaGymaaqaaiaaiMdaaaGaeyOeI0IaaGOnaiabg2da 9maalaaabaGaeyOeI0IaaGymaiaaiMdaaeaacaaI5aaaaiabgkHiTm aalaaabaGaaGOnaaqaaiaaigdaaaaaaa@4178@                           

[L.C.M. of 9 and 1 is 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaabc cacaqGHbGaaeOBaiaabsgacaqGGaGaaGymaiaabccacaqGPbGaae4C aiaabccacaaI5aaaaa@3F43@  ]

= 19×1 9×1 6×9 1×9 = 19 9 54 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaigdacaaI5aGaey41aqRaaGymaaqaaiaa iMdacqGHxdaTcaaIXaaaaiabgkHiTmaalaaabaGaaGOnaiabgEna0k aaiMdaaeaacaaIXaGaey41aqRaaGyoaaaaaeaacqGH9aqpdaWcaaqa aiabgkHiTiaaigdacaaI5aaabaGaaGyoaaaacqGHsisldaWcaaqaai aaiwdacaaI0aaabaGaaGyoaaaaaaaa@4F74@

= 1954 9 = 73 9 =8 1 9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaigdacaaI5aGaeyOeI0IaaGynaiaaisda aeaacaaI5aaaaaqaaiabg2da9maalaaabaGaeyOeI0IaaG4naiaaio daaeaacaaI5aaaaaqaaiabg2da9iabgkHiTiaaiIdadaWcaaqaaiaa igdaaeaacaaI5aaaaaaaaa@4515@

Question: 3

Find the product:

(i)             9 2 ×( 7 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aaabaGaaGOmaaaacqGHxdaTdaqadaqaamaalaaabaGaeyOeI0Ia aG4naaqaaiaaisdaaaaacaGLOaGaayzkaaaaaa@3D83@

(ii)          3 10 ×(9) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaGymaiaaicdaaaGaey41aqRaaiikaiabgkHiTiaaiMda caGGPaaaaa@3D3A@

(iii)      6 5 × 9 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI2aaabaGaaGynaaaacqGHxdaTdaWcaaqaaiaaiMdaaeaa caaIXaGaaGymaaaaaaa@3CB4@

(iv)       3 7 ×( 2 5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaG4naaaacqGHxdaTdaqadaqaamaalaaabaGaeyOeI0Ia aGOmaaqaaiaaiwdaaaaacaGLOaGaayzkaaaaaa@3D7E@

(v)          3 11 × 2 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaGymaiaaigdaaaGaey41aq7aaSaaaeaacaaIYaaabaGa aGynaaaaaaa@3BBD@

(vi)       3 5 × 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaeyOeI0IaaGynaaaacqGHxdaTdaWcaaqaaiabgkHiTiaa iwdaaeaacaaIZaaaaaaa@3CE1@

Solution

(i)             9 2 ×( 7 4 ) = 9×(7) 2×4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiMdaaeaacaaIYaaaaiabgEna0oaabmaabaWaaSaaaeaacqGH sislcaaI3aaabaGaaGinaaaaaiaawIcacaGLPaaaaeaacqGH9aqpda WcaaqaaiaaiMdacqGHxdaTcaGGOaGaeyOeI0IaaG4naiaacMcaaeaa caaIYaGaey41aqRaaGinaaaaaaaa@4811@

        = 63 8 =7 7 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaiAdacaaIZaaabaGaaGioaaaaaeaacqGH 9aqpcqGHsislcaaI3aWaaSaaaeaacaaI3aaabaGaaGioaaaaaaaa@3E68@

(ii)          3 10 ×(9) = 3×(9) 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiodaaeaacaaIXaGaaGimaaaacqGHxdaTcaGGOaGaeyOeI0Ia aGyoaiaacMcaaeaacqGH9aqpdaWcaaqaaiaaiodacqGHxdaTcaGGOa GaeyOeI0IaaGyoaiaacMcaaeaacaaIXaGaaGimaaaaaaaa@45A8@

        = 27 10 =2 7 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiabgkHiTiaaikdacaaI3aaabaGaaGymaiaaicdaaaaa baGaeyypa0JaeyOeI0IaaGOmamaalaaabaGaaG4naaqaaiaaigdaca aIWaaaaaaaaa@3FC9@

(iii)      6 5 × 9 11 = (6)×9 5×11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaiAdaaeaacaaI1aaaaiabgEna0oaalaaabaGaaGyo aaqaaiaaigdacaaIXaaaaaqaaiabg2da9maalaaabaGaaiikaiabgk HiTiaaiAdacaGGPaGaey41aqRaaGyoaaqaaiaaiwdacqGHxdaTcaaI XaGaaGymaaaaaaaa@47FC@

        = 54 55 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacqGHsislcaaI1aGaaGinaaqaaiaaiwdacaaI1aaaaaaa@3AD6@

(iv)       3 7 ×( 2 5 ) = 3×(2) 7×5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiodaaeaacaaI3aaaaiabgEna0oaabmaabaWaaSaaaeaacqGH sislcaaIYaaabaGaaGynaaaaaiaawIcacaGLPaaaaeaacqGH9aqpda WcaaqaaiaaiodacqGHxdaTcaGGOaGaeyOeI0IaaGOmaiaacMcaaeaa caaI3aGaey41aqRaaGynaaaaaaaa@4807@

        = 6 35 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacqGHsislcaaI2aaabaGaaG4maiaaiwdaaaaaaa@3A17@

(v)          3 11 × 2 5 = 3×2 11×5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiodaaeaacaaIXaGaaGymaaaacqGHxdaTdaWcaaqaaiaaikda aeaacaaI1aaaaaqaaiabg2da9maalaaabaGaaG4maiabgEna0kaaik daaeaacaaIXaGaaGymaiabgEna0kaaiwdaaaaaaaa@44B5@

        = 6 55 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI2aaabaGaaGynaiaaiwdaaaaaaa@392C@

(vi)       3 5 ×( 5 3 ) = 3×(5) 5×3 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiodaaeaacqGHsislcaaI1aaaaiabgEna0oaabmaabaWaaSaa aeaacqGHsislcaaI1aaabaGaaG4maaaaaiaawIcacaGLPaaaaeaacq GH9aqpdaWcaaqaaiaaiodacqGHxdaTcaGGOaGaeyOeI0IaaGynaiaa cMcaaeaacqGHsislcaaI1aGaey41aqRaaG4maaaacqGH9aqpcaaIXa aaaaa@4BA0@

Question: 4

Find the value of:

(i)             (4)÷ 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgk HiTiaaisdacaGGPaGaey49aG7aaSaaaeaacaaIYaaabaGaaG4maaaa aaa@3CA0@

(ii)          3 5 ÷2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIZaaabaGaaGynaaaacqGH3daUcaaIYaaaaa@3B48@  

(iii)      4 5 ÷(3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaacqGH3daUcaGGOaGaeyOeI0IaaG4m aiaacMcaaaa@3D90@  

(iv)       1 8 ÷ 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIXaaabaGaaGioaaaacqGH3daUdaWcaaqaaiaaiodaaeaa caaI0aaaaaaa@3C18@  

(v)          2 13 ÷ 1 7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaIYaaabaGaaGymaiaaiodaaaGaey49aG7aaSaaaeaacaaI XaaabaGaaG4naaaaaaa@3CD0@  

(vi)       7 12 ÷ 2 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI3aaabaGaaGymaiaaikdaaaGaey49aG7aaSaaaeaacqGH sislcaaIYaaabaGaaGymaiaaiodaaaaaaa@3E79@  

(vii)    3 13 ÷ 4 65 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaGymaiaaiodaaaGaey49aG7aaSaaaeaacqGHsislcaaI 0aaabaGaaGOnaiaaiwdaaaaaaa@3D92@  

Solution

(i)             (4)÷ 2 3 =(4)× 3 2 =(2)×3 =6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaGGOa GaeyOeI0IaaGinaiaacMcacqGH3daUdaWcaaqaaiaaikdaaeaacaaI Zaaaaaqaaiabg2da9iaacIcacqGHsislcaaI0aGaaiykaiabgEna0o aalaaabaGaaG4maaqaaiaaikdaaaaabaGaeyypa0JaaiikaiabgkHi TiaaikdacaGGPaGaey41aqRaaG4maaqaaiabg2da9iabgkHiTiaaiA daaaaa@4DE1@

(ii)           

3 5 ÷2 = 3 5 × 1 2 = (3)×1 5×2 = 3 10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaiodaaeaacaaI1aaaaiabgEpa4kaaikdaaeaacqGH 9aqpdaWcaaqaaiabgkHiTiaaiodaaeaacaaI1aaaaiabgEna0oaala aabaGaaGymaaqaaiaaikdaaaaabaGaeyypa0ZaaSaaaeaacaGGOaGa eyOeI0IaaG4maiaacMcacqGHxdaTcaaIXaaabaGaaGynaiabgEna0k aaikdaaaaabaGaeyypa0ZaaSaaaeaacqGHsislcaaIZaaabaGaaGym aiaaicdaaaaaaaa@511F@  

 

(iii)       

4 5 ÷(3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHsislcaaI0aaabaGaaGynaaaacqGH3daUcaGGOaGaeyOeI0IaaG4m aiaacMcaaaa@3D90@

= (4) 5 × 1 (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaGGOaGaeyOeI0IaaGinaiaacMcaaeaacaaI1aaaaiabgEna 0oaalaaabaGaaGymaaqaaiaacIcacqGHsislcaaIZaGaaiykaaaaaa a@4096@

= (4)×1 5×(3) = 4 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaacIcacqGHsislcaaI0aGaaiykaiabgEna0kaaigda aeaacaaI1aGaey41aqRaaiikaiabgkHiTiaaiodacaGGPaaaaaqaai abg2da9maalaaabaGaaGinaaqaaiaaigdacaaI1aaaaaaaaa@45F1@  

(iv)        

1 8 ÷ 3 4 = 1 8 × 4 3 = (1)×1 2×3 = 1 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiabgkHiTiaaigdaaeaacaaI4aaaaiabgEpa4oaalaaabaGaaG4m aaqaaiaaisdaaaaabaGaeyypa0ZaaSaaaeaacqGHsislcaaIXaaaba GaaGioaaaacqGHxdaTdaWcaaqaaiaaisdaaeaacaaIZaaaaaqaaiab g2da9maalaaabaGaaiikaiabgkHiTiaaigdacaGGPaGaey41aqRaaG ymaaqaaiaaikdacqGHxdaTcaaIZaaaaaqaaiabg2da9maalaaabaGa eyOeI0IaaGymaaqaaiaaiAdaaaaaaaa@5139@  

 

(v)           

2 13 ÷ 1 7 = 2 13 × 7 1 = (2)×7 13×1 = 14 13 =1 1 13 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaaikdaaeaacaaIXaGaaG4maaaacqGH3daUdaWcaaqa aiaaigdaaeaacaaI3aaaaaqaaiabg2da9maalaaabaGaeyOeI0IaaG OmaaqaaiaaigdacaaIZaaaaiabgEna0oaalaaabaGaaG4naaqaaiaa igdaaaaabaGaeyypa0ZaaSaaaeaacaGGOaGaeyOeI0IaaGOmaiaacM cacqGHxdaTcaaI3aaabaGaaGymaiaaiodacqGHxdaTcaaIXaaaaaqa aiabg2da9maalaaabaGaeyOeI0IaaGymaiaaisdaaeaacaaIXaGaaG 4maaaaaeaacqGH9aqpcqGHsislcaaIXaWaaSaaaeaacaaIXaaabaGa aGymaiaaiodaaaaaaaa@59D3@

(vi)        

7 12 ÷ 2 13 = 7 12 × 13 (2) = (7)×13 12×(2) = 91 24 =3 19 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkHiTiaaiEdaaeaacaaIXaGaaGOmaaaacqGH3daUdaWcaaqa aiabgkHiTiaaikdaaeaacaaIXaGaaG4maaaacqGH9aqpdaWcaaqaai abgkHiTiaaiEdaaeaacaaIXaGaaGOmaaaacqGHxdaTdaWcaaqaaiaa igdacaaIZaaabaGaaiikaiabgkHiTiaaikdacaGGPaaaaaqaaiabg2 da9maalaaabaGaaiikaiabgkHiTiaaiEdacaGGPaGaey41aqRaaGym aiaaiodaaeaacaaIXaGaaGOmaiabgEna0kaacIcacqGHsislcaaIYa GaaiykaaaaaeaacqGH9aqpdaWcaaqaaiabgkHiTiaaiMdacaaIXaaa baGaeyOeI0IaaGOmaiaaisdaaaaabaGaeyypa0JaaG4mamaalaaaba GaaGymaiaaiMdaaeaacaaIYaGaaGinaaaaaaaa@624D@

(vii)     

         3 13 ÷ 4 65 = 3 13 × 65 (4) = 3×(5) 1×4 = 15 4 =3 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiodaaeaacaaIXaGaaG4maaaacqGH3daUdaWcaaqaaiabgkHi TiaaisdaaeaacaaI2aGaaGynaaaaaeaacqGH9aqpdaWcaaqaaiaaio daaeaacaaIXaGaaG4maaaacqGHxdaTdaWcaaqaaiaaiAdacaaI1aaa baGaaiikaiabgkHiTiaaisdacaGGPaaaaaqaaiabg2da9maalaaaba GaaG4maiabgEna0kaacIcacqGHsislcaaI1aGaaiykaaqaaiaaigda cqGHxdaTcaaI0aaaaaqaaiabg2da9maalaaabaGaeyOeI0IaaGymai aaiwdaaeaacaaI0aaaaaqaaiabg2da9iabgkHiTiaaiodadaWcaaqa aiaaiodaaeaacaaI0aaaaaaaaa@5A85@