NCERT Solutions for Class 8 Maths Chapter 11 Direct and Inverse Proportions
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Page 1
NCERT
SOLUTIONS
CLASS - 8TH
aglase .co
SOLUTIONS
CLASS - 8TH
aglase .co
Page 2
Book : Mathematics Ncert Solutions | Chapter-13 Maths
Class : 8th
Subject : Maths
Chapter : 13
Chapter Name : Direct and Inverse Proportions
Exercise 13.1
Q1 Following are the car parking charges near a railway station upto
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.
Answer. 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 2 12 3 24 2
As each ratio is not same, therefore, the parking charges are not in a direct proportion to the parking time.
Q2 A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. In the following table, nd the parts of base that
need to be added.
Parts of red pigment 1 4 7 12 20
Parts of base 8 … … …
Answer. 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.
Therefore, the parts of red pigments and the parts Of base are in direct proportion.
Parts of red pigment 1 4 7 12 20
Parts of base 8 x1 x2 x3
According to direct proportion ,
x1 8
= ⇒ x1 = 4 × 8 = 32
4 1
x2 8
= ⇒ x2 = 7 × 8 = 56
7 1
x3 8
= ⇒ x3 = 8 × 12 = 96
12 1
x4 8
= ⇒ x4 = 8 × 20 = 160
20 1
The Table can be drawn as follows.
Parts of red pigment 1 4 7 12 20
Parts of base 8 32 56 96 160
Q3 In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how much red pigment should we mix with 1800 mL of base?
Answer. 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
Page 1 of 7 Aglasem Schools
Class : 8th
Subject : Maths
Chapter : 13
Chapter Name : Direct and Inverse Proportions
Exercise 13.1
Q1 Following are the car parking charges near a railway station upto
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.
Answer. 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 2 12 3 24 2
As each ratio is not same, therefore, the parking charges are not in a direct proportion to the parking time.
Q2 A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. In the following table, nd the parts of base that
need to be added.
Parts of red pigment 1 4 7 12 20
Parts of base 8 … … …
Answer. 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.
Therefore, the parts of red pigments and the parts Of base are in direct proportion.
Parts of red pigment 1 4 7 12 20
Parts of base 8 x1 x2 x3
According to direct proportion ,
x1 8
= ⇒ x1 = 4 × 8 = 32
4 1
x2 8
= ⇒ x2 = 7 × 8 = 56
7 1
x3 8
= ⇒ x3 = 8 × 12 = 96
12 1
x4 8
= ⇒ x4 = 8 × 20 = 160
20 1
The Table can be drawn as follows.
Parts of red pigment 1 4 7 12 20
Parts of base 8 32 56 96 160
Q3 In Question 2 above, if 1 part of a red pigment requires 75 mL of base, how much red pigment should we mix with 1800 mL of base?
Answer. 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
Page 1 of 7 Aglasem Schools
Page 3
Book : Mathematics Ncert Solutions | Chapter-13 Maths
1 x
=
75 1800
1×1800
⇒ x =
75
⇒ x = 24
Thus, 24 parts of red pigments should be mixed with 1800 mL of base.
Q4 A machine in a soft drink factory lls 840 bottles in six hours. How many bottles will it ll in ve hours?
Answer. Let the number of bottles lled by the machine in ve 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 ll these bottles are in direct proportion.
Therefore, we obtain
840 x
=
6 5
840×5
x = = 700
6
Thus, 700 bottles will be lled in 5 hours.
Q5 A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the
bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?
Answer. Let the actual length of bacteria be x cm and the enlarged length of bacteria be y cm. if the photograph is enlarged for 20,000 times.
The number of times the photograph of bacteria was enlarged and the length of bacteria are in direct proportion.
Therefore, we obtain
5 x
=
50,000 1
1 −4
⇒ x = = 10
10000
Hence, the actual length of bacteria is 10 cm.
−4
Let the length Of bacteria when the photograph Of bacteria is enlarged 20, 000 times be y.
5 y
=
50,000 20,000
20,000×5
y = = 2
50,000
Hence, the enlarged length of bacteria is 2 cm.
Q6 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?
Answer. 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 9cm ×
Actual ship 12m 28m
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
x = = 21
12
Thus, the length of the model ship is 21 cm.
Q7 Suppose 2 kg of sugar contains 9 × 106 crystals. How many sugar crystals are there in (i) 5 kg of sugar? (ii) 1.2 kg of sugar?
Answer. (i) Let the number of sugar crystals in 5 kg of sugar be x.
Page 2 of 7 Aglasem Schools
1 x
=
75 1800
1×1800
⇒ x =
75
⇒ x = 24
Thus, 24 parts of red pigments should be mixed with 1800 mL of base.
Q4 A machine in a soft drink factory lls 840 bottles in six hours. How many bottles will it ll in ve hours?
Answer. Let the number of bottles lled by the machine in ve 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 ll these bottles are in direct proportion.
Therefore, we obtain
840 x
=
6 5
840×5
x = = 700
6
Thus, 700 bottles will be lled in 5 hours.
Q5 A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the
bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?
Answer. Let the actual length of bacteria be x cm and the enlarged length of bacteria be y cm. if the photograph is enlarged for 20,000 times.
The number of times the photograph of bacteria was enlarged and the length of bacteria are in direct proportion.
Therefore, we obtain
5 x
=
50,000 1
1 −4
⇒ x = = 10
10000
Hence, the actual length of bacteria is 10 cm.
−4
Let the length Of bacteria when the photograph Of bacteria is enlarged 20, 000 times be y.
5 y
=
50,000 20,000
20,000×5
y = = 2
50,000
Hence, the enlarged length of bacteria is 2 cm.
Q6 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?
Answer. 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 9cm ×
Actual ship 12m 28m
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
x = = 21
12
Thus, the length of the model ship is 21 cm.
Q7 Suppose 2 kg of sugar contains 9 × 106 crystals. How many sugar crystals are there in (i) 5 kg of sugar? (ii) 1.2 kg of sugar?
Answer. (i) Let the number of sugar crystals in 5 kg of sugar be x.
Page 2 of 7 Aglasem Schools
Page 4
Book : Mathematics Ncert Solutions | Chapter-13 Maths
Amount of sugar (in kg) 2 5
6
Number of crystals 9 × 10 x
The amount of sugar and the number of crystals it contains are directly proportional to each other. Therefore, we obtain
2 5
6
=
x
9×10
6
5×9×10 7
x = = 2.25 × 10
2
Hence, the number Of sugar crystals is 2.25 x 10 . 7
(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
6
Number of crystals 9 × 10 y
2 1.2
=
6 y
9×10
6
1.2×9×10 6
y = = 5.4 × 10
2
Hence, the number of sugar crystals is 5.4 × 10 6
Q8 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?
Answer. Let the distance represented on the map be x cm.
Distance covered on road in (in km) 18 72
Distance represented on map (in cm) 1 x
The distances covered on road and represented on map are directly proportional toeach other. Therefore, we obtain
18 72
=
1 x
72
⇒ x = = 4
18
Hence, the distance represented on the map is 4 cm.
Q9 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 5m long.
Answer. (i) Let the length of the shadow of the other pole be x m.
1 m = 100 cm
The given information in the form Of a table is as follows.
Height of pole (in m) 5.60 10.50
Length of shadow (in m) 3.20 x
More the height of an object, more will be the length of its shadow.
Thus, 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
⇒ x = = 6
5.60
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
Length of shadow (in m) 3.20 5
The height of the pole and the length of the shadow are directly proportional to each other. Therefore,
5.60 y
=
3.20 5
5×5.60
y = = 8.75
3.20
Thus , the height of the pole is 8.75 m or 8 m 75 cm.
Q10 A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?
Page 3 of 7 Aglasem Schools
Amount of sugar (in kg) 2 5
6
Number of crystals 9 × 10 x
The amount of sugar and the number of crystals it contains are directly proportional to each other. Therefore, we obtain
2 5
6
=
x
9×10
6
5×9×10 7
x = = 2.25 × 10
2
Hence, the number Of sugar crystals is 2.25 x 10 . 7
(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
6
Number of crystals 9 × 10 y
2 1.2
=
6 y
9×10
6
1.2×9×10 6
y = = 5.4 × 10
2
Hence, the number of sugar crystals is 5.4 × 10 6
Q8 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?
Answer. Let the distance represented on the map be x cm.
Distance covered on road in (in km) 18 72
Distance represented on map (in cm) 1 x
The distances covered on road and represented on map are directly proportional toeach other. Therefore, we obtain
18 72
=
1 x
72
⇒ x = = 4
18
Hence, the distance represented on the map is 4 cm.
Q9 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 5m long.
Answer. (i) Let the length of the shadow of the other pole be x m.
1 m = 100 cm
The given information in the form Of a table is as follows.
Height of pole (in m) 5.60 10.50
Length of shadow (in m) 3.20 x
More the height of an object, more will be the length of its shadow.
Thus, 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
⇒ x = = 6
5.60
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
Length of shadow (in m) 3.20 5
The height of the pole and the length of the shadow are directly proportional to each other. Therefore,
5.60 y
=
3.20 5
5×5.60
y = = 8.75
3.20
Thus , the height of the pole is 8.75 m or 8 m 75 cm.
Q10 A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?
Page 3 of 7 Aglasem Schools
Page 5
Book : Mathematics Ncert Solutions | Chapter-13 Maths
Answer. Let the distance travelled by the truck in 5 hours be x km.
We know, 1 hour = 60 minutes
Therefore , 5 hours = (5 x 60) minutes = 300 minutes
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 x
=
25 300
14×300
x = = 168
25
Hence, the distance travelled by the truck is 168 km.
Exercise 13.2
Q1 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 in a uniform speed.
(iii) Area of cultivated land and the crop harvested.
(iv) The time taken for a xed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.
Answer. (i) These are in inverse proportion because if there are more workers, then it will take lesser time to complete that job.
(ii) NO, these are not in inverse proportion because in more time, we may cover more distance with a uniform speed.
(iii) No, these are not in inverse proportion because in more area, more quantity of crop may be harvested.
(iv) These are in inverse proportion because with more speed, we may complete a certain distance in a lesser time.
(v) These are in inverse proportion because if the population is increasing, then the area of the land per person will be decreasing
accordingly.
Page : 213 , Block Name : Exercise 13.2
Q2 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
nd whether the prize money given to an individual winner is directly or inversely proportional to the number of winners?
Number of winners 1 1 2 4 5 10
Prize for each winner (in z) 1, 00, 000 50, 000 … … …
Answer. A table of the given information is as follows.
Number of winners 1 2 4 10
Prize for each winner (in Rs) 100000 50000 x1 x2 x3 x5
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,
100000 100000 100000
1 × 100000 = 4 × x1 => x1 = = 250001 × 10000 = 5 × x2 => x2 = = 200001 × 100000 = 8 × x3 x3 = = 12500
4 5 8
100000 100000
1 × 100000 = 10 × x4 => x4 = = 100001 × 10000 = 20 × x5 => x5 = = 5000
10 20
Page : 213 , Block Name : Exercise 13.2
Q3 Rehman is making a wheel using spokes. He wants to x equal spokes in such a way that the angles between any pair of consecutive
spokes are equal. Help him by completing the following table.
Page 4 of 7 Aglasem Schools
Answer. Let the distance travelled by the truck in 5 hours be x km.
We know, 1 hour = 60 minutes
Therefore , 5 hours = (5 x 60) minutes = 300 minutes
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 x
=
25 300
14×300
x = = 168
25
Hence, the distance travelled by the truck is 168 km.
Exercise 13.2
Q1 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 in a uniform speed.
(iii) Area of cultivated land and the crop harvested.
(iv) The time taken for a xed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.
Answer. (i) These are in inverse proportion because if there are more workers, then it will take lesser time to complete that job.
(ii) NO, these are not in inverse proportion because in more time, we may cover more distance with a uniform speed.
(iii) No, these are not in inverse proportion because in more area, more quantity of crop may be harvested.
(iv) These are in inverse proportion because with more speed, we may complete a certain distance in a lesser time.
(v) These are in inverse proportion because if the population is increasing, then the area of the land per person will be decreasing
accordingly.
Page : 213 , Block Name : Exercise 13.2
Q2 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
nd whether the prize money given to an individual winner is directly or inversely proportional to the number of winners?
Number of winners 1 1 2 4 5 10
Prize for each winner (in z) 1, 00, 000 50, 000 … … …
Answer. A table of the given information is as follows.
Number of winners 1 2 4 10
Prize for each winner (in Rs) 100000 50000 x1 x2 x3 x5
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,
100000 100000 100000
1 × 100000 = 4 × x1 => x1 = = 250001 × 10000 = 5 × x2 => x2 = = 200001 × 100000 = 8 × x3 x3 = = 12500
4 5 8
100000 100000
1 × 100000 = 10 × x4 => x4 = = 100001 × 10000 = 20 × x5 => x5 = = 5000
10 20
Page : 213 , Block Name : Exercise 13.2
Q3 Rehman is making a wheel using spokes. He wants to x equal spokes in such a way that the angles between any pair of consecutive
spokes are equal. Help him by completing the following table.
Page 4 of 7 Aglasem Schools
Page 6
Book : Mathematics Ncert Solutions | Chapter-13 Maths
(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°?
Answer. A table of the given information is as follows.
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,
∘
∘ 4×90 ∘
4 × 90 = x1 × 8 => x1 = = 45
8
∘ ∘
Similarly, x 2 =
4×90
10
= 36
∘
x3 =
4×90 ∘
= 30
12
Thus, the following table is obtained.
(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 ∘
4 × 90 = 15 × x => x = = 24
15
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 40 angles between a pair of consecutive spokes, be y.
∘
Therefore,
∘ ∘ 4×90
4 × 90 = y × 40 => y = = 9
40
Hence, the number of spokes in such a wheel is 9.
Page : 213 , Block Name : Exercise 13.2
Q4 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?
Answer. Number of remaining children = 24 — 4 = 20
Let the number Of sweets which each Of the 20 students will get, be x.
The following table is obtained.
If the number of students is lesser, then each student will get more number of sweets.
Since this is a case of inverse proportion,
24×5
24 × 5 = 20 × x => x = = 6
20
Hence, each student will get 6 sweets.
Page : 214 , Block Name : Exercise 13.2
Q5 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?
Answer. 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.
Page 5 of 7 Aglasem Schools
(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°?
Answer. A table of the given information is as follows.
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,
∘
∘ 4×90 ∘
4 × 90 = x1 × 8 => x1 = = 45
8
∘ ∘
Similarly, x 2 =
4×90
10
= 36
∘
x3 =
4×90 ∘
= 30
12
Thus, the following table is obtained.
(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 ∘
4 × 90 = 15 × x => x = = 24
15
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 40 angles between a pair of consecutive spokes, be y.
∘
Therefore,
∘ ∘ 4×90
4 × 90 = y × 40 => y = = 9
40
Hence, the number of spokes in such a wheel is 9.
Page : 213 , Block Name : Exercise 13.2
Q4 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?
Answer. Number of remaining children = 24 — 4 = 20
Let the number Of sweets which each Of the 20 students will get, be x.
The following table is obtained.
If the number of students is lesser, then each student will get more number of sweets.
Since this is a case of inverse proportion,
24×5
24 × 5 = 20 × x => x = = 6
20
Hence, each student will get 6 sweets.
Page : 214 , Block Name : Exercise 13.2
Q5 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?
Answer. 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.
Page 5 of 7 Aglasem Schools
Page 7
Book : Mathematics Ncert Solutions | Chapter-13 Maths
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
20 × 6 = 30 × x => x = = 4
30
Thus, the food will last for 4 days.
Page : 214 , Block Name : Exercise 13.2
Q6 A contractor estimates that 3 persons could rewire Jasminder’s house in 4 days. If, he uses 4 persons instead of three, how long should
they take to complete the job?
Answer. Let the number Of days required by 4 persons to complete the job be x.
The following table is obtained.
Number of days 4 x
Number of persons 3 4
If the number of persons is more, then it will take lesser time to complete the job.
Hence, the number of days and the number of persons required to complete the job are inversely proportional to each other.
Therefore,
4×3
4 × 3 = x × 4x = = 3
4
Thus, the number of days required to complete the job is 3.
Page : 214 , Block Name : Exercise 13.2
Q7 A batch of bottles were 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 lled?
Answer. Let the number Of boxes lled, 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
12 × 25 = 20 × x => x = = 15
20
Hence, the number of boxes required to pack these bottles is 15.
Page : 214 , Block Name : Exercise 13.2
Q8 A factory requires 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?
Answer. 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
42 × 63 = 54 × x => x = = 49
54
Hence, the required number of machines to produce the given number of articles in 54 days is 49.
Page : 214 , Block Name : Exercise 13.2
Q9 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?
Answer. Let the time taken by the car to reach the destination, while travelling with a speed of 80 km/hr, be x hours.
The following table is obtained.
Page 6 of 7 Aglasem Schools
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
20 × 6 = 30 × x => x = = 4
30
Thus, the food will last for 4 days.
Page : 214 , Block Name : Exercise 13.2
Q6 A contractor estimates that 3 persons could rewire Jasminder’s house in 4 days. If, he uses 4 persons instead of three, how long should
they take to complete the job?
Answer. Let the number Of days required by 4 persons to complete the job be x.
The following table is obtained.
Number of days 4 x
Number of persons 3 4
If the number of persons is more, then it will take lesser time to complete the job.
Hence, the number of days and the number of persons required to complete the job are inversely proportional to each other.
Therefore,
4×3
4 × 3 = x × 4x = = 3
4
Thus, the number of days required to complete the job is 3.
Page : 214 , Block Name : Exercise 13.2
Q7 A batch of bottles were 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 lled?
Answer. Let the number Of boxes lled, 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
12 × 25 = 20 × x => x = = 15
20
Hence, the number of boxes required to pack these bottles is 15.
Page : 214 , Block Name : Exercise 13.2
Q8 A factory requires 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?
Answer. 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
42 × 63 = 54 × x => x = = 49
54
Hence, the required number of machines to produce the given number of articles in 54 days is 49.
Page : 214 , Block Name : Exercise 13.2
Q9 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?
Answer. Let the time taken by the car to reach the destination, while travelling with a speed of 80 km/hr, be x hours.
The following table is obtained.
Page 6 of 7 Aglasem Schools
Page 8
Book : Mathematics Ncert Solutions | Chapter-13 Maths
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 3 1
60 × 2 = 80 × x => x = = = 1
80 2 2
The time required by the car to reach the given destination is 1 hours.
1
2
Page : 214 , Block Name : Exercise 13.2
Q10 Two persons could t new windows in a house in 3 days.
(i) One of the persons fell ill before the work started. How long would the job take now?
(ii) How many persons would be needed to t the windows in one day?
Answer. (i) Let the number of days required by 1 man to t all the windows be x. The
following table is obtained.
Lesser the number of persons, more will be the number of days required to t all the
windows. Hence, this is a case of inverse proportion. Therefore,
2 × 3 = 1 × x => x = 6
Hence, the number Of days taken by 1 man to t all the windows is 6.
(ii) Let the number of persons required to t all the windows in one day be y. The following table is formed.
Lesser the number Of days, more will be the number Of persons required to t all the
windows. Hence, this is a case of inverse proportion. Therefore,
2 × 3 = y × 1 => y = 6
Hence, 6 persons are required to t all the windows in one day.
Page : 214 , Block Name : Exercise 13.2
Q11 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?
Answer. Let the duration of each period, when there are 9 periods a day in the school, be x minutes. The following table is obtained.
If there is 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
45 × 8 = x × 9 => x = = 40
9
Hence, in this case, the duration of each period will be 40 minutes.
Page : 214 , Block Name : Exercise 13.2
Page 7 of 7 Aglasem Schools
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 3 1
60 × 2 = 80 × x => x = = = 1
80 2 2
The time required by the car to reach the given destination is 1 hours.
1
2
Page : 214 , Block Name : Exercise 13.2
Q10 Two persons could t new windows in a house in 3 days.
(i) One of the persons fell ill before the work started. How long would the job take now?
(ii) How many persons would be needed to t the windows in one day?
Answer. (i) Let the number of days required by 1 man to t all the windows be x. The
following table is obtained.
Lesser the number of persons, more will be the number of days required to t all the
windows. Hence, this is a case of inverse proportion. Therefore,
2 × 3 = 1 × x => x = 6
Hence, the number Of days taken by 1 man to t all the windows is 6.
(ii) Let the number of persons required to t all the windows in one day be y. The following table is formed.
Lesser the number Of days, more will be the number Of persons required to t all the
windows. Hence, this is a case of inverse proportion. Therefore,
2 × 3 = y × 1 => y = 6
Hence, 6 persons are required to t all the windows in one day.
Page : 214 , Block Name : Exercise 13.2
Q11 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?
Answer. Let the duration of each period, when there are 9 periods a day in the school, be x minutes. The following table is obtained.
If there is 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
45 × 8 = x × 9 => x = = 40
9
Hence, in this case, the duration of each period will be 40 minutes.
Page : 214 , Block Name : Exercise 13.2
Page 7 of 7 Aglasem Schools
Document Details
| Board / Org | NCERT |
| Exam | Class 8 |
| Type | Solution |
| Pages | 8 |
| Language | english |
| Updated | 30 Apr 2026 |