NCERT Solutions Class 8 Maths Chapter 11 Direct and Inverse Proportions Exercise 11.2
NCERT Solutions for Class 8 Maths Chapter 11 – Direct and Inverse Proportions Ex 11.2 are given here in simple, step-by-step form. NCERT stands for National Council of Educational Research and Training, the board that makes your textbook. Chapter 11 teaches how two quantities change together. In Exercise 11.1 you learnt direct proportion, where both quantities go up or come down together. Exercise 11.2 teaches the opposite idea, called inverse proportion. Here, when one quantity goes up, the other comes down.
This exercise has 11 questions. They are all from daily life – prize money shared between winners, spokes in a wheel, sweets shared among children, food for animals, workers on a job, speed and time, and school periods. Every question is solved below in short steps, with the answer written in bold at the end. Students can use it for homework and revision, parents can use it to check the work at home, and teachers can use it in class. A free printable PDF of these solutions is also available, so you can study even without the internet.
NCERT Solutions Class 8 Maths Chapter 11 Direct and Inverse Proportions Exercise 11.2
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| Number of spokes | 4 | 6 | 8 | 10 | 12 |
| Angle betweem a pair of consecutive spokes | 90º | 60º | … | … | … |
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4 hours ₹ 60
8 hours ₹ 100
12 hours ₹ 140
24 hours ₹ 180
Check if the parking charges are in direct proportion to the parking time.
Exercise 11.2 – Questions and Answers
Q 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.
Answer:
(i) It is in inverse proportion, because if there are more workers, then it will take less 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) It is in inverse proportion, because with more speed we may complete a certain distance in a lesser time.
(v) It is in inverse proportion, because if the population is increasing/decreasing, then the area of the land per person will be decreasing/increasing accordingly.
Q 2. In a Television game show, the prize money of ₹ 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 | 8 | 10 | 20 |
|---|---|---|---|---|---|---|---|
| Prize of each winner (in ₹) | 1,00,000 | 50,000 | … | … | … | … | … |
Answer:
Let the unknown prizes be x₁, x₂, x₃, x₄ and x₅.
From the table, we obtain
1 × 1,00,000 = 2 × 50,000 = 1,00,000
Thus, the number of winners and the amount given to each winner are inversely proportional to each other.
Therefore,
1 × 1,00,000 = 4 × x₁ ⇒ x₁ = 1,00,000/4 = 25,000
1 × 1,00,000 = 5 × x₂ ⇒ x₂ = 1,00,000/5 = 20,000
1 × 1,00,000 = 8 × x₃ ⇒ x₃ = 1,00,000/8 = 12,500
1 × 1,00,000 = 10 × x₄ ⇒ x₄ = 1,00,000/10 = 10,000
1 × 1,00,000 = 20 × x₅ ⇒ x₅ = 1,00,000/20 = 5,000
The completed table is:
| Number of winners | 1 | 2 | 4 | 5 | 8 | 10 | 20 |
|---|---|---|---|---|---|---|---|
| Prize of each winner (in ₹) | 1,00,000 | 50,000 | 25,000 | 20,000 | 12,500 | 10,000 | 5,000 |
Hence, the prize money given to an individual winner is inversely proportional to the number of winners.
Q 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°?
Answer:
Let the unknown angles be x₁, x₂ and x₃.
From the given table, we obtain
4 × 90° = 360° = 6 × 60°
(i) Thus, the number of spokes and the angle between a pair of consecutive spokes are inversely proportional to each other.
Now we will find x₁, x₂ and x₃.
4 × 90° = 8 × x₁ ⇒ x₁ = (4 × 90°)/8 = 45°
x₂ = (4 × 90°)/10 = 36°
x₃ = (4 × 90°)/12 = 30°
Thus, the following table is obtained:
| Number of spokes | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|
| Angle between a pair of consecutive spokes | 90° | 60° | 45° | 36° | 30° |
(ii) Let the angle between a pair of consecutive spokes on a wheel with 15 spokes be x.
Therefore, 4 × 90° = 15 × x ⇒ x = (4 × 90°)/15 = 24°
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° = y × 40° ⇒ y = (4 × 90°)/40° = 9
Hence, the number of spokes in such a wheel is 9.
Q 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?
Answer:
Total number of children = 24
If the number of the children is reduced by 4, the number of remaining children = 24 – 4 = 20
Let the number of sweets which each of the 20 students will get be x.
The given information is represented in the following table:
| Number of students | 24 | 20 |
|---|---|---|
| Number of sweets | 5 | x |
If the number of children is reduced, then each student will get more number of sweets. So, this is the case of inverse proportion.
Therefore, we obtain
24 × 5 = 20 × x
⇒ x = (24 × 5)/20 = 6
Hence, each student will get 6 sweets.
Q 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?
Answer:
Total number of animals is 20.
If there are 10 more animals, then the total number of animals would be 30.
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 | 30 |
|---|---|---|
| Number of days | 6 | x |
The food will last longer if there is a smaller number of animals.
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
⇒ x = (20 × 6)/30 = 4
Therefore, the food will last for 4 days.
Q 6. 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 given information is represented in the following table:
| Number of days | 4 | x |
|---|---|---|
| Number of persons | 3 | 4 |
If more persons are employed, 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 = x × 4
⇒ x = (4 × 3)/4 = 3
Therefore, the number of days required by 4 persons to complete the job is 3.
Q 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?
Answer:
Let the number of boxes filled by using 20 bottles in each box be x.
The given information is represented in the following table:
| Number of bottles | 12 | 20 |
|---|---|---|
| Number of boxes | 25 | x |
If a greater number of bottles are used, there will be a smaller 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
⇒ x = (12 × 25)/20 = 15
Hence, the number of boxes required to pack 20 bottles in each box is 15.
Q 8. 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 the same number of articles in 54 days be x.
The given information is represented in the following table:
| Number of machines | 42 | x |
|---|---|---|
| Number of days | 63 | 54 |
If a greater number of machines is used, then a smaller number of days it will take to produce the given number of articles.
So, the number of machines and the number of days required to produce the given number of articles are inversely proportional to each other.
Therefore,
42 × 63 = 54 × x
⇒ x = (42 × 63)/54 = 49
Hence, the required number of machines to produce the given number of articles in 54 days is 49.
Q 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?
Answer:
Let the time taken by the car to reach the destination, when it travels at a speed of 80 km/h, be x hours.
The given information is represented in the following table:
| Speed (in km/h) | 60 | 80 |
|---|---|---|
| Time taken (in hours) | 2 | x |
If the speed of the car is more, then it will take less time 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
⇒ x = (60 × 2)/80 = 3/2
Therefore, the time required by the car to reach the given destination is 3/2 hours, that is, 1½ hours.
Q 10. Two persons could fit 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 fit the windows in one day?
Answer:
(i) Since one of the persons fell ill before the work started, we are left with only one person.
Let the number of days required by 1 man to fit all the windows be x.
| Number of persons | 2 | 1 |
|---|---|---|
| Number of days | 3 | x |
If a smaller number of persons are employed, then it will take more number of days to fit 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 fit all the windows is 6.
(ii) Let the number of persons required to fit all the windows in one day be y.
| Number of persons | 2 | y |
|---|---|---|
| Number of days | 3 | 1 |
If a smaller number of days is given, then more people would be required to fit all the windows.
Hence, it is a case of inverse proportion.
Therefore,
2 × 3 = y × 1
y = 6
Hence, 6 persons are required to fit all the windows in one day.
Q 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?
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 from the given information:
| Duration of each period (in minutes) | 45 | x |
|---|---|---|
| Number of periods | 8 | 9 |
If there is a greater number of periods a day in the school, then the duration of each period will be lesser.
Hence, it is a case of inverse proportion.
Therefore,
45 × 8 = x × 9
⇒ x = (45 × 8)/9 = 40
Hence, the duration of each period, when there are 9 periods a day in the school, will be 40 minutes.
Related Links
- NCERT Solutions Class 8 Maths Chapter 11 Exercise 11.1
- NCERT Solutions Class 8 Maths Chapter 11 – Direct and Inverse Proportions
- NCERT Solutions Class 8 Maths Chapter 10 – Exponents and Powers
- NCERT Solutions Class 8 Maths Chapter 12 – Factorisation
- NCERT Solutions Class 8 Maths Chapter 7 – Comparing Quantities
- NCERT Solutions Class 8 Maths – All Chapters
- NCERT Solutions Class 8 – All Subjects