Unit 4: Simple Equations

# Question: 1

Complete the last column of the table.

 S. No. Equation Value Say, whether the equation is satisfied. (Yes/ No) (i) $x+3=0$ $x=3$ (ii) $x+3=0$ $x=0$ (iii) $x+3=0$ $x=-3$ (iv) $x–7=1$ $x=7$ (v) $x–7=1$ $x=8$ (vi) $5x=25$ $x=0$ (vii) $5x=25$ $x=5$ (viii) $5x=25$ $x=-5$ (ix) $\frac{m}{3}=2$ $m=-6$ (x) $\frac{m}{3}=2$ $m=0$ (xi) $\frac{m}{3}=2$ $m=6$

## Solution

 S. No. Equation Value Say, whether the equation is satisfied. (Yes/ No) (i) $x+3=0$ $x=3$ No (ii) $x+3=0$ $x=0$ No (iii) $x+3=0$ $x=-3$ Yes (iv) $x–7=1$ $x=7$ No (v) $x–7=1$ $x=8$ Yes (vi) $5x=25$ $x=0$ No (vii) $5x=25$ $x=5$ Yes (viii) $5x=25$ $x=-5$ No (ix) $\frac{m}{3}=2$ $m=-6$ No (x) $\frac{m}{3}=2$ $m=0$ No (xi) $\frac{m}{3}=2$ $m=6$ Yes

# Question: 2

Check whether the value given in the brackets is a solution to the given equation or not:

a.    $n+5=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=1\right)$

b.   $7n+5=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=–2\right)$

c.    $7n+5=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=2\right)$

d.   $4p–3=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p=1\right)$

e.    $4p–3=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p=–4\right)$

f.     $4p–3=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p=0\right)$

## Solution

a.    $n+5=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=1\right)$

Putting

is not the solution of given equation.

b.   $7n+5=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=–2\right)$

Putting

is not the solution of given equation.

c.    $7n+5=19\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n=2\right)$

Putting

is the solution of given equation.

d.   $4p–3=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p=1\right)$

Putting

is not the solution of given equation.

e.    $4p–3=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p=–4\right)$

Putting

is not the solution of given equation.

f.     $4p–3=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(p=0\right)$

Putting

is not the solution of given equation.

# Question: 3

Solve the following equations by trial and error method:

(i)             $5p+2=17$

(ii)          $3m–14=4$

## Solution

(i)             $5p+2=17$

Putting

is not the solution.

Putting

is not the solution.

Putting

is the solution.

(ii)          $3m–14=4$

Putting

is not the solution.

Putting

is not the solution.

Putting

is not the solution.

Putting

is not the solution.

Putting

is not the solution.

Putting

is the solution.

# Question: 4

Write equations for the following statements:

(i)             The sum of numbers $x$ and $4$ is $9$.

(ii)          The difference between $y$ and $2$ is $8$.

(iii)      Ten times $a$ is $70$.

(iv)       The number $b$ divided by $5$ gives $6$.

(v)          Three fourth of $t$ is $15$.

(vi)       Seven times $m$ plus $7$ gets you $77$.

(vii)    One fourth of a number minus $4$ gives $4$.

(viii)If you take away $6$ from $6$ times $y$, you get $60$.

(ix)       If you add $3$ to one third of $z$, you get $30$.

## Solution

(i)             $x+4=9$

(ii)          $y-2=8$

(iii)      $10a=70$

(iv)       $\frac{b}{5}=6$

(v)          $\frac{3}{4}t=15$

(vi)       $7m+7=77$

(vii)    $\frac{x}{4}-4=4$

(viii)$6y-6=60$

(ix)       $\frac{z}{3}+3=30$

# Question: 5

Write the following equations in statement forms:

(i)             $p+4=15$

(ii)          $m–7=3$

(iii)      $2m=7$

(iv)       $\frac{m}{5}=3$

(v)          $\frac{3m}{5}=6$

(vi)       $3p+4=25$

(vii)    $4p–2=18$

(viii) $\frac{p}{2}+2=8$

## Solution

(i)             The sum of numbers

(ii)          7 subtracted from

(iii)      Two times

(iv)       The number $m$ is divided by $5$ gives $3$.

(v)          Three-fifth of the number $m$ is $6$

(vi)       Three times $p$ plus $4$ gives $25$.

(vii)    If you take away $2$ from $4$ times $p$, you get $18$.

(viii) If you add $2$ to the half of $p$, you get $8$.

# Question: 6

Set up an equation in the following cases:

(i)             Irfan says that he has $7$ marbles more than five times the marbles Parmit has. Irfan has $37$ marbles. (Take $m$ to be the number of Parmit’s marbles.)

(ii)          Laxmi’s father is $49$ years old. He is $4$ years older than three times Laxmi’s age. (Take Laxmi’s age to be $y$ years.)

(iii)      The teacher tells the class that the highest marks obtained by a student in her class is twice the lowest marks plus $7$. The highest score is $87$. (Take the lowest score to be $l$.)

(iv)       In an isosceles triangle, the vertex angle is twice either base angle. (Let the base angle be $b$ in degrees. Remember that the sum of angles of a triangle is $180$ degrees).

## Solution

a.    Let $m$ be the number of Parmit’s marbles.

b.   Let the age of Laxmi be $y$ years.

c.    Let the lowest score be $l$.

d.   Let the base angle of the isosceles triangle be $b$, so vertex angle $=2b$

[Angle sum property of a $\Delta$ ]

# Question: 1

Give first the step you will use to separate the variable and then solve the equation:

a.    $x–1=0$

b.   $x+1=0$

c.    $x-1=5$

d.   $x+6=2$

e.    $y-4=-7$

f.     $y-4=4$

g.   $y+4=4$

h.   $y+4=–4$

## Solution

a.    $x–1=0$

[Adding $1$ to both sides]

b.   $x+1=0$

[Subtracting $1$ from both sides]

c.    $x-1=5$

[Adding $1$ to both sides]

d.   $x+6=2$

[Subtracting $6$ from both sides]

e.    $y-4=-7$

[Adding $4$ to both sides]

f.     $y-4=4$

[Adding $4$ to both sides]

g.   $y+4=4$

[Subtracting $4$ from both sides]

h.   $y+4=–4$

[Subtracting $4$ from both sides]

# Question: 2

Give first the step you will use to separate the variable and then solve the equation:

a.    $3l=42$

b.   $\frac{b}{2}=6$

c.    $\frac{p}{7}=4$

d.   $4x=25$

e.    $8y=36$

f.     $\frac{z}{3}=\frac{5}{4}$

g.   $\frac{a}{5}=\frac{7}{15}$

h.   $20t=-10$

## Solution

a.    $3l=42$

[Dividing both sides by $3$ ]

b.   $\frac{b}{2}=6$

[Multiplying both sides by $2$ ]

c.    $\frac{p}{7}=4$

[Multiplying both sides by $7$ ]

d.   $4x=25$

[Dividing both sides by $4$ ]

e.    $8y=36$

[Dividing both sides by $8$ ]

f.     $\frac{z}{3}=\frac{5}{4}$

[Multiplying both sides by $3$ ]

g.   $\frac{a}{5}=\frac{7}{15}$

[Multiplying both sides by $5$ ]

h.   $20t=-10$

[Dividing both sides by $20$ ]

# Question: 3

Give the steps you will use to separate the variable and then solve the equation:

a.    $3n-2=46$

b.   $5m+7=17$

c.    $\frac{20p}{3}=40$

d.   $\frac{3p}{10}=6$

## Solution

a.    $3n-2=46$

Step I: $3n-2+2=46+2$

$⇒3n=48$                   [Adding $2$ to both sides]

Step II: $\frac{3n}{3}=\frac{48}{3}⇒n=16$     [Dividing both sides by $3$ ]

b.   $5m+7=17$

Step I: $5m+7-7=17-7$

$⇒5m=10$                  [Subtracting $7$ from both sides]

Step II: $\frac{5m}{5}=\frac{10}{5}⇒m=2$    [Dividing both sides by $5$ ]

c.    $\frac{20p}{3}=40$

Step I: $\frac{20p}{3}×3=40×3$

$⇒20p=120$               [Multiplying both sides by $3$ ]

Step II: $\frac{20p}{20}=\frac{120}{20}⇒p=6$   [Dividing both sides by $20$ ]

d.   $\frac{3p}{10}=6$

Step I: $\frac{3p}{10}×10=6×10$

$⇒3p=60$                   [Multiplying both sides by $10$ ]

Step II: $\frac{3p}{3}=\frac{60}{3}⇒p=20$     [Dividing both sides by $3$ ]

# Question: 4

Solve the following equations:

a.    $10p=100$

b.   $10p+10=100$

c.    $\frac{p}{4}=5$

d.   $\frac{-p}{3}=5$

e.    $\frac{3p}{4}=6$

f.     $3s=-9$

g.   $3s+12=0$

h.   $3s=0$

i.      $2q=6$

j.      $2q-6=0$

k.   $2q+6=0$

l.      $2q+6=12$

## Solution

a.    $10p=100$

[Dividing both sides by $10$ ]

b.   $10p+10=100$

[Subtracting $10$ from both sides ]

[Dividing both sides by $10$ ]

c.    $\frac{p}{4}=5$

[Multiplying both sides by $4$ ]

d.   $\frac{-p}{3}=5$

[Multiplying both sides by $-3$ ]

e.    $\frac{3p}{4}=6$

[Multiplying both sides by $4$ ]

[Dividing both sides by $3$ ]

f.     $3s=-9$

[Dividing both sides by $3$ ]

g.   $3s+12=0$

[Subtracting $12$ from both sides]

[Dividing both sides by $3$ ]

h.   $3s=0$

[Dividing both sides by $3$ ]

i.      $2q=6$

[Dividing both sides by $2$ ]

j.      $2q-6=0$

[Adding $6$ to both sides]

[Dividing both sides by $2$ ]

k.   $2q+6=0$

[Subtracting $6$ from both sides]

[Dividing both sides by $2$ ]

l.      $2q+6=12$

[Subtracting $6$ from both sides]

[Dividing both sides by $2$ ]

# Question: 1

Solve the following equations.

a.    $2y+\frac{5}{2}=\frac{37}{2}$

b.   $5t+28=10$

c.    $\frac{a}{5}+3=2$

d.   $\frac{q}{4}+7=5$

e.    $\frac{5}{2}x=10$

f.     $\frac{5}{2}x=\frac{25}{4}$

g.   $7m+\frac{19}{2}=13$

h.   $6z+10=-2$

i.      $\frac{3l}{2}=\frac{2}{3}$

j.      $\frac{2b}{3}-5=3$

## Solution

a.    $2y+\frac{5}{2}=\frac{37}{2}$

b.   $5t+28=10$

c.    $\frac{a}{5}+3=2$

d.   $\frac{q}{4}+7=5$

e.    $\frac{5}{2}x=10$

f.     $\frac{5}{2}x=\frac{25}{4}$

g.   $7m+\frac{19}{2}=13$

h.   $6z+10=-2$

i.      $\frac{3l}{2}=\frac{2}{3}$

j.      $\frac{2b}{3}-5=3$

# Question: 2

Solve the following equations.

a.    $2\left(x+4\right)=12$

b.   $3\left(n-5\right)=21$

c.    $3\left(n-5\right)=-21$

d.   $3-2\left(2-y\right)=7$

e.    $-4\left(2-x\right)=9$

f.     $4\left(2-x\right)=9$

g.   $4+5\left(p-1\right)=34$

h.   $34-5\left(p-1\right)=4$

## Solution

a.    $2\left(x+4\right)=12$

b.   $3\left(n-5\right)=21$

c.    $3\left(n-5\right)=-21$

d.   $3-2\left(2-y\right)=7$

e.    $-4\left(2-x\right)=9$

f.     $4\left(2-x\right)=9$

g.   $4+5\left(p-1\right)=34$

h.   $34-5\left(p-1\right)=4$

# Question: 3

Solve the following equations.

a.    $4=5\left(p-2\right)$

b.   $-4=5\left(p-2\right)$

c.    $–16=-5\left(2-p\right)$

d.   $10=4+3\left(t+2\right)$

e.    $28=4+3\left(t+5\right)$

f.     $0=16+4\left(m-6\right)$

## Solution

a.    $4=5\left(p-2\right)$

b.   $-4=5\left(p-2\right)$

c.    $-16=-5\left(2-p\right)$

d.   $10=4+3\left(t+2\right)$

e.    $28=4+3\left(t+5\right)$

f.     $0=16+4\left(m-6\right)$

# Question: 4

a.    Construct $3$ equations starting with $x=2$

b.   Construct $3$ equations starting with $x=-2$

## Solution

a.    $3$ equations starting with $x=2$.

(i) $x=2$

Multiplying both sides by $10$, $10x=20$

Adding $2$ to both sides

$10x+2=20+2=10x+2=22$

(ii)          $x=2$

Multiplying both sides by $5$,           $5x=10$

Subtracting $3$ from both sides

$5x-3=10-3⇒5x-3=7$

(iii)      $x=2$

Dividing both sides by $5$,                $\frac{x}{5}=\frac{2}{5}$

b.   $3$ equations starting with $x=-2$.

(i) $x=-2$

Multiplying both sides by $3$,           $3x=-6$

(ii)          $x=-2$

Multiplying both sides by $3$,           $3x=-6$

Adding $7$ to both sides

$3x+7=-6+7⇒3x+7=1$

(iii)      $x=-2$

Multiplying both sides by $3$,           $3x=-6$

Adding $10$ to both sides

$3x+10=-6+10=3x+10=4$

# Question: 1

Set up equations and solve them to find the unknown numbers in the following cases:

a.    Add $4$ to eight times a number; you get $60$.

b.   One fifth of a number minus $4$ gives $3$.

c.    If I take three fourths of a number and count up $3$ more, I get $21$.

d.   When I subtracted $11$ from twice a number, the result was $15$.

e.    Munna subtracts thrice the number of notebooks he has from $50$, he finds the result to be $8$.

f.     Ibenhal thinks of a number. If she adds $19$ to it and divides the sum by $5$, she will get $8$.

g.   Anwar thinks of a number. If he takes away $7$ from $\frac{5}{2}$ of the number, the result is $\frac{11}{2}$.

## Solution

a.    Let the number be $x$.

According to the question, $8x+4=60$

b.   Let the number be $y$.

According to the question, $\frac{y}{5}-4=3$

c.    Let the number be $z$.

According to the question, $\frac{3}{4}z+3=21$

d.   Let the number be $x$.

According to the question, $2x-11=15$

e.    Let the number be $m$.

According to the question, $50-3m=8$

f.     Let the number be $n$.

According to the question, $\frac{n+19}{5}=8$

g.   Let the number be $x$.

According to the question, $\frac{5}{2}x-7=\frac{11}{2}$

# Question: 2

Solve the following:

a.    The teacher tells the class that the highest marks obtained by a student in her class is twice the lowest marks plus $7$. The highest score is $87$. What is the lowest score?

b.   In an isosceles triangle, the base angles are equal. The vertex angle is $40°$. What are the base angles of the triangle? (Remember, the sum of three angles of a triangle is $180°$ ).

c.    Smita’s mother is $34$ years old. Two years from now mother’s age will be $4$ times Smita’s present age. What is Smita’s present age?

d.   Sachin scored twice as many runs as Rahul. Together, their runs fell two short of a double century. How many runs did each one score?

## Solution

a.    Let the lowest marks be $y$.

According to the question, $2y+7=87$

Thus, the lowest score is $40$.

b.   Let the base angle of the triangle be $b$.

Given, $a=40°,\text{\hspace{0.17em}}b=c$

Since, $a+b+c=180°$     [Angle sum property of a triangle]

Thus, the base angles of the isosceles triangle are $70°$ each.

c.    Let Smita’s present age be $x$

$2$ years from now, Smita’s age $=x+2$

$2$ years from now her mother’s age $=4\left(x+2\right)$

i.e., $36=4\left(x+2\right)$

$\left(x+2\right)=\frac{36}{4}=9$

Smita’s present age $=7$ years.

d.   Let the score of Rahul be $x$ runs and Sachin’s score is $2x$.

According to the question, $x+2x=198$

Thus, Rahul’s score $=66$ runs

And Sachin’s score $=2×66=132$ runs.

# Question: 3

Solve the following:

(i)             Irfan says that he has $7$ marbles more than five times the marbles Parmit has. Irfan has $37$ marbles. How many marbles does Parmit have?

(ii)          Laxmi's father is $49$ years old. He is $4$ years older than three times Laxmi's age. What is Laxmi's age?

(iii)      Maya, Madhura and Mohsina are friends studying in the same class. In a class test in geography, Maya got $16$ out of $25$. Madhura got $20$. Their average score was $19$. How much did Mohsina score?

(iv)       People of Sundargram planted a total of $102$ trees in the village garden. Some of the trees were fruit trees. The number of non-fruit trees were two more than three times the number of fruit trees. What was the number of fruit trees planted?

## Solution

(i)             Let the number of marbles Parmit has be $m$.

According to the question, $5m+7=37$

Thus, Parmit has $6$ marbles.

(ii)          Let the age of Laxmi be $y$ year.

Then her father’s age $=\left(3y+4\right)$ years

According to question, $3y+4=49$

Thus, the age of Laxmi is $15$ years.

(iii)      Let the marks scored by Mohsina $=x$

According to the question scored by Maya $=16$

According to the question scored by Madhura $=20$

Total marks scored by all three $=16+20+x=36+x$

Average marks scored by all three $=19$

Total marks scored by all three $=16+20+x=36+x$

$36+x=57$

Transposing $36$ will make it $-36$.

$x=57-36$

$x=21$

Hence the marks scored by Mohsina are $21$ out of $25$.

(iv)       Let the number of fruit trees be $t$.

Then the number of non-fruits tree $=3t+2$

According to the question, $t+3t+2=102$

Thus, the number of fruit trees are $25$.

# Question: 4

Solve the following riddle:

I am a number,

Tell my identity!

Take me seven times over

Let the number be $n$.
According to the question, $7n+50+40=300$
Thus, the required number is $30.$