 Lesson: Direct and Inverse Proportions
Exercise 13.1
Question 1
The car parking charges near a railway station up to.
4 hours Rs 60
8 hours Rs 100
12 hours Rs 140
24 hours Rs 180
Check if the parking charges are in direct proportion to the parking time.
No
Solution:
A table of the given information is formed as
Number of hours
4
8
12
24
Parking charges (in Rs)
60
100
140
180
The ratio of parking charges to the respective number of hours (Rs/ hour)
can be calculated as
60 100 25 140 35 180 15
15, , ,
4 8 4 12 3 24 2
As each ratio is not the same, therefore, the parking charges are not in a
direct proportion to the parking time.
Question 2
A mixture of paint is prepared by mixing 1 part of red pigment
with 8 parts of base.
In the following table, find the parts of base that need to be added.
Parts of red pigment
1
7
12
20
parts of base
8
……
…….
…… 32, 56, 96, 160
Solution:
The given mixture of paint is prepared by mixing 1 part of red pigments
with 8 parts of base.
For more parts of red pigments, the parts of the base will also be more.
Also, the parts of red pigments and the parts of base should be in direct proportion.
The given information in the form of a table is as follows.
Parts of red pigment
1
4
7
12
20
parts of base
8
1
x
2
x
3
x
4
x
According to direct proportion,
1
8
41
x

1
4 8 32x
2
8
71
x

1
7 8 56x
3
8
12 1
x

1
12 8 96x
4
8
20 1
x

1
20 8 160x
Parts of red pigment
1
4
7
12
20
Parts of base
8
32
56
96
160
Question 3
If 1 part of a red pigment requires 75 ml of base, how much red pigment
should we mix with 1800 ml of base?
24
Solution:
Let the parts of red pigment required to mix with 1800 mL of base be x.
The given information in the form of a table is as follows.
Parts of red pigment
1
x
Parts of base (in ml)
75
1800 The parts of red pigment and the parts of base are in direct proportion.
Therefore, we obtain
1
75 1800
x
1 1800
75
x

24x
Thus, 24 parts of red pigments should be mixed with 1800 ml of base.
Question 4
A machine in a soft drink factory fills 840 bottles in six hours.
How many bottles will it fill in five hours?
700
Solution:
Let the number of bottles filled by the machine in five hours be x.
The given information in the form of a table is as follows.
Number of bottles
840
x
Time taken (in hours)
6
5
The number of bottles and the time taken to fill these bottles are
in direct proportion.
Therefore, we obtain
840
65
x
840 5
700
6
x

Thus, 700 bottles will be filled in 5 hours.
Question 5
A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm.
What is the actual length of the bacteria?
If the photograph is enlarged 20,000 times only,
What would be its enlarged length? 4
10 m, 2 m
Solution:
Let the actual length of bacteria be x cm and the enlarged length of bacteria
be y cm, if the photograph is enlarged 20,000 times.
The given information in the form of a table is as follows.
Length of the bacteria (in cm)
5
x
y
Number of times the photograph of
the bacteria was enlarged
50000
1
20000
The number of times the photograph of bacteria was enlarged and the length
of bacteria are in direct proportion.
Therefore, we obtain
5
50,000 1
x
4
1
10
10000
x
Hence, the actual length of the bacteria is
4
10
cm.
Let the length of the bacteria when the photograph of the bacteria is enlarged
20, 000 times be y.
5
50,000 20,000
y

20,000 5
2
50,000
y
Hence, the enlarged length of the bacteria is 2 cm.
Question 6
In a model of a ship, the mast is 9 cm high,
while the mast of the actual ship is 12 m high.
If the length of the ship is 28 m, how long is the model ship? 21 m
Solution:
Let the length of the mast of the model ship be x cm.
The given information in the form of a table is as follows:
---
Height of mast
Length of ship
Model ship
9 cm
x
Actual ship
12 m
28 m
We know that the dimensions of the actual ship and the model ship
are directly proportional to each other.
Therefore, we obtain:
12 28
9 x
28 9
21
12
x
Thus, the length of the model ship is 21 m.
Question 7
Suppose 2 kg of sugar contains
6
9 10
crystals.
How many sugar crystals are there in?
(i) 5 kg of sugar?
(ii) 1.2 kg of sugar?
7
(i) 2.25 10
,
6
(ii) 5.4 10
Solution:
(i) Let the number of sugar crystals in 5 kg of sugar be x.
The given information in the form of a table is as follows.
Amount in kg
2
5
Number of crystals
6
9 10
x
The amount of sugar and the number of crystals it contains
are directly proportional to each other.
Therefore, we obtain
6
25
9 10 x
6
7
5 9 10
2.25 10
2
x

Hence, the number of sugar crystals is
7
2.25 10
(ii) Let the number of sugar crystals in 1.2 kg of sugar be y.
The given information in the form of a table is as follows. Amount of sugar (in kg)
2
1.2
Number of crystals
6
9 10
y
6
2 1.2
9 10 y
6
6
1.2 9 10
5.4 10
2
y

Hence, the number of sugar crystals is
6
5.4 10
Question 8
Rashmi has a road map with a scale of 1 cm representing 18 km.
She drives on a road for 72 km.
What would be her distance covered in the map?
4 cm
Solution:
Let the distance represented on the map be x cm.
The given information in the form of a table is as follows.
Distance covered on road in (in km)
18
72
Distance represented on map (in cm)
1
x
18 72
1 x
72
4
18
x
Hence, the distance represented on the map is 4 cm.
Question 9
A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long.
Find at the same time
(i) the length of the shadow cast by another pole 10 m 50 cm high
(ii) the height of a pole which casts a shadow 5 m long.
(i) 6 m (ii) 8 m 75 cm. Solution:
(i) Let the length of the shadow of the other pole be x m
The given information in the form of a table is as follows.
Height of the pole (in m)
5.60
10.50
Length of the shadow (in m)
3.20
x
More is the height of an object; more will be the length of its shadow.
The height of an object and length of its shadow are directly proportional
to each other.
Therefore, we obtain
5.60 10.50
3.20 x
10.50 3.20
6
5.60
x
Hence, the length of the shadow will be 6 m.
(ii) Let the height of the pole be y m.
The given information in the form of a table is as follows.
Height of pole (in m)
5.60
y
3.20
5
The height of the pole and the length of the shadow are directly
proportional to each other.
Therefore,
5.60
3.20 5
y
5 5.60
8.75
3.20
y
Thus, the height of the pole is 8.75 m or 8 m 75 cm.
Question 10
A loaded truck travels 14 km in 25 minutes.
If the speed remains the same,
how far can it travel in 5 hours?
168 km
Solution:
Let the distance travelled by the truck in 5 hours be x km.
We know, 1 hour = 60 minutes
5 hours
5 60
minutes
300
minutes
The given information in the form of a table is as follows.
Distance travelled (in km) 14 x
Time (in min) 25 300
The distance travelled by the truck and the time taken by the truck are directly proportional to each other.
Therefore,
14
25 300
x
14 300
168
25
x
Hence, the distance travelled by the truck is 168 km.
Exercise 13.2
Question 1
Which of the following are in inverse proportion?
(i) The number of workers on a job and the time to complete the job.
(ii) The time taken for a journey and the distance travelled at a uniform speed.
(iii) Area of cultivated land and the crop harvested.
(iv) The time taken for a fixed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.
(i), (iv), (v)
Solution:
(i) This is the case of inverse proportion because if there are more workers,
then it will take lesser time to complete that job.
(ii) This is not the case of inverse proportion because in more time,
more distance will be covered at a uniform speed.
(iii) This is not the case of inverse proportion because in more area of cultivated land,
more quantity of crop may be harvested.
(iv) This is the case of inverse proportion because with more speed,
a certain distance will be covered in a lesser time.
(v) This is the case of inverse proportion because if the population is increasing,
then the area of the land per person will be decreasing consequently.
Question 2
In a television game show, the prize money of Rs 1, 00,000 is to be
divided equally amongst the winners.
Complete the following table and find whether the prize money given
to an individual winner is directly or inversely proportional to the number of winners?
Number
of
winners
1
2
4
5
6
10
20
Prize for
each
winner
(in Rs)
100000
50000
-------
-------
-------
-------
-
------- 25000, 20000, 12500, 10000, 5000
Solution:
A table of the given information is as follows.
Number
of
winners
1
2
4
5
6
10
20
Prize for
each
winner
(in Rs)
100000
50000
1
x
2
x
3
x
4
x
5
x
From the table, we obtain
1 100000 2 50000 100000
Thus, the number of winners and the amount given to each winner are inversely
proportional to each other.
Therefore,
1
1 100000 4 x
1
100000
25000
4
x
2
1 100000 5 x
2
100000
20000
5
x
3
1 100000 8 x
3
100000
12500
8
x
4
1 100000 10 x
4
100000
10000
5
x
5
1 100000 20 x
5
100000
5000
20
x
Question 3
Rehman is making a wheel using spokes.
He wants to fix equal spokes in such a way that the angles between
any pair of consecutive spokes are equal.
Help him by completing the following table. Number of spokes
4
6
8
10
12
Angle between a pair of consecutive spokes
90
60
-----
-----
-----
(i) Are the number of spokes and the angles formed between the pairs of
consecutive spokes in inverse proportion?
(ii) Calculate the angle between a pair of consecutive spokes on a wheel
with 15 spokes.
(iii) How many spokes would be needed, if the angle between a pair of
consecutive spokes is
40
?
45 ,36 ,30
(i) Yes (ii)
24
(iii) 9
Solution:
A table of the given information is as follows.
Number of spokes
4
6
8
10
12
Angle between a pair of consecutive spokes
90
60
1
x
2
x
3
x
From the given table, we obtain
4 90 360 6 60    
Thus, the number of spokes and the angle between a pair of consecutive
spokes are inversely proportional to each other.
Therefore,
1
4 90 8x
1
4 90
45
8
x
2
4 90
36
10
x
And
3
4 90
30
12
x Number of spokes
4
6
8
10
12
Angle between a pair of consecutive spokes
90
60
45
36
30
(i) Yes, the number of spokes and the angles formed between the pairs of
consecutive spokes are in inverse proportion.
(ii) Let the angle between a pair of consecutive spokes on a wheel with 15
spokes be
x
.
Therefore,
4 90 15 x
4 90
24
15
x
Hence, the angle between a pair of consecutive spokes of a wheel,
which has 15 spokes in it, is
24
.
(iii) Let the number of spokes in a wheel, which has 40º angles between
a pair of consecutive spokes, be
y
.
Therefore,
4 90 40y
4 90
9
40
y

Hence, the number of spokes in such a wheel is 9.
Question 4
If a box of sweets is divided among 24 children, they will get 5 sweets each.
How many would each get, if the number of the children is reduced by 4?
6
Solution:
Remaning number of children
24 4 20
Let the number of sweets which each of the 20 children will get, be x.
The following table is obtained.
Number of children
24
20
Number of sweets
5
x
If the number of children is lesser, then each child will get more
number of sweets.
Since this is a case of inverse proportion,
24 5 20 x
24 5
6
20
x
Hence, each child will get 6 sweets. Question 5
A farmer has enough food to feed 20 animals in his cattle for 6 days.
How long would the food last if there were 10 more animals in his cattle?
4 days
Solution:
Let the number of days that the food will last if there were
10 more animals in the cattle be x.
The following table is obtained.
Number of animals
20
20 10 30
Number of days
6
x
More the number of animals,
lesser will be the number of days for which the food will last.
Hence, the number of days the food will last and the number of animals
are inversely proportional to each other.
Therefore,
20 6 30 x
20 6
4
30
x
Thus, the food will last for 4 days
Question 6
A contractor estimates that 3 people could rewire Jasminder’s house in 4 days.
If he uses 4 people instead of three,
how long should they take to complete the job?
3 days
Solution:
Let the number of days required by 4 people to complete the job be x.
The following table is obtained.
Number of days
4
x
Number of people
3
4
If the number of people is more, then it will take lesser
time to complete the job.
Hence, the number of days and the number of people required
to complete the job are inversely proportional to each other.
Therefore,
4 3 4x
43
3
4
x Thus, the number of days required to complete the job is 3.
Question 7
A batch of bottles was packed in 25 boxes with 12 bottles in each box.
If the same batch is packed using 20 bottles in each box,
how many boxes would be filled?
15 boxes
Solution:
Let the number of boxes filled, by using 20 bottles in each box be x.
The following table is obtained.
Number of bottles
12
20
Number of boxes
25
x
More the number of bottles, lesser will be the number of boxes.
Hence, the number of bottles and the number of boxes required to
pack these are inversely proportional to each other.
Therefore,
12 25 20 x
12 25
15
20
x
Hence, 15 boxes will be filled if 20 bottles are packed in each.
Question 8
A factory required 42 machines to produce a given number
of articles in 63 days.
How many machines would be required to produce the same
number of articles in 54 days?
49 machines Solution:
Let the number of machines required to produce articles in 54 days be x.
The following table is obtained.
Number of machines
42
x
Number of days
63
54
More the number of machines,
lesser will be the number of days that it will take to produce the
given number of articles.
Thus, this is a case of inverse proportion.
Therefore,
42 63 54 x
42 63
49
54
x
Hence, 49 machines are required to produce articles in 54 days.
Question 9
A car takes 2 hours to reach a destination by travelling at the speed
of 60 km/h.
How long will it take when the car travels at the speed of 80 km/h?
1
1
2
Hours
Solution:
Let the time taken by the car to reach the destination,
While travelling at a speed of 80 km/hr, be x hours.
The following table is obtained.
Speed (in km/hr)
60
80
Time taken (in hours)
2
x
More the speed of the car,
lesser will be the time taken by it to reach the destination.
Hence, the speed of the car and the time taken by the car are
inversely proportional to each other.
Therefore,
60 2 80 x
60 2 3 1
1
80 2 2
x
The time required by the car to reach the given destination is
1
1
2
hours Question 10
Two people could fit new windows in house in 3 days.
(i) One of the people fell ill before the work started.
How long would the job take now?
(ii) How many people would be needed to fit the windows in one day?
(a) days (ii) 6persons (i) Let the number of days required by 1 man to
fit all the windows be x.
The following table is obtained.
Solution:
Number of persons
2
1
Number of days
3
x
Lesser the number of people,
more will be the number of days required to fit all the windows.
Hence, this is a case of inverse proportion.
Therefore,
2 3 1 x
6x
Hence, the number of days taken by 1 man to fit all the windows is 6.
(ii) Let the number of people required to fit all the windows in one day be y.
The following table is formed.
Number of persons
2
y
Number of days
3
1
Lesser the number of days,
more will be the number of people required to fit all the windows.
Hence, this is a case of inverse proportion.
Therefore,
2 3 1 y
6y
Hence, 6 people are required to fit all the windows in one day.
Question 11
A school has 8 periods a day each of 45 minutes duration.
How long would each period be, if the school has 9 periods a day,
Assuming the number of school hours to be the same?
40 minutes
Solution:
Let the duration of each period,
When there are 9 periods a day in the school,
Be x minutes.
The following table is obtained. Duration of each period (in minutes)
45
x
Number of periods
8
9
If there are more number of periods a day in the school,
Then the duration of each period will be lesser.
Hence, this is a case of inverse proportion.
Therefore
45 8 9x
45 8
40
9
x
Hence, in this case, the duration of each period will be 40 minutes.