Exercise 11.2

1. Which of the following are in inverse proportion?

Instructions

The number of workers on a job and the time to complete the job.

The time taken for a journey and the distance travelled in a uniform speed.

Area of cultivated land and the crop harvested.

The time taken for a fixed journey and the speed of the vehicle.

The population of a country and the area of land per person.

Directly Proportional

Inversely Proportional

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	Prize for each winner (in Rs.)
1	1,00,000
2	50,000
4	...
5	...
8	...
10	...
20	...

Instructions

Here, the number of winners and prize money are in proportion because winners are increasing, and prize money is .

When the number of winners is 4, each winner will get = 1000004 = Rs.

When the number of winners is 5, each winner will get = 1000005 = Rs.

When the number of winners is 8, each winner will get = 1000008 = Rs.

When the number of winners is 10, each winner will get = 10000010 = Rs.

When the number of winners is 20, each winner will get = 10000020 = Rs.

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	Angle between a pair of consecutive spokes
4	90°
6	60°
8	...
10	...
12	...

(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°?

Instructions

Here, the number of spokes is and the angle between a pair of consecutive spokes is .

So, it is an proportion, and the angle at the centre of a circle is degree.

When the number of spokes is 8, then the angle between a pair of consecutive spokes = 3608 = degree

When the number of spokes is 10, then the angle between a pair of consecutive spokes = 36010 = degree.

When the number of spokes is 12, then the angle between a pair of consecutive spokes = 36012 = degree.

(i) , the number of spokes and the angles formed between a pair of consecutive spokes are in proportion.

(ii) When the number of spokes is 15, then the angle between a pair of consecutive spokes = 36015 = degree.

(iii) The number of spokes would be needed = 36040 =

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?

Instructions

Each child gets sweets. So, 24 children will get sweets.

So, Total number of sweets = 120

If the number of children is reduced by 4, then children left = - =

Now, each child will get sweets = 12020 = sweets

Thus, each child now gets 6 sweets.

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?

Instructions

Let the number of days be x. We also have: Total number of animals = 20 + =

Here, the number of animals and the number of days are in proportion.

So, 2030 = x6 which gives: = × 6

Further: x = 20×630 i.e. x =

Hence, the food will last for four days.

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?

Instructions

Let the time taken to complete the job be x. Here, the number of persons and the number of days are in proportion.

So, 34 = which gives: 3 × =

We get: x = 3×44 i.e. x =

Hence, 4 persons will complete the job in 3 days.

7. 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 filled?

Instructions

Let the number of boxes be x. Here, the number of bottles and the number of boxes are in proportion.

So, 1220 = which gives: 12 × =

We get: x = 12×2520 =

Hence, 15 boxes would be filled.

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?

Instructions

Let the number of machines required be x. Here, the number of machines and the number of days are in proportion.

This gives us: 6354 = which simplifies to: × 42 =

So, x = 63×4254 =

Hence, 49 machines would be required.

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?

Instructions

Let the number of hours be x. Here, the speed of the car and time are in proportion.

So, 6080 = which gives: × 2 =

Thus, x = 60×280 = which means x = hours and minutes.

Hence, the car will take 1 12 hr to reach the destination.

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?

Instructions

(i) Let the number of days be x. Here, the number of persons and the number of days are in proportion.

So, 21 = which gives: x = days

(ii) Let the number of persons be x.

Again, the number of persons and the number of days are in inverse proportion. So, 2x = 13 which gives x = persons.

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?

Instructions

Let the duration of each period be x.

Here, the number of periods and the duration of periods are in proportion.

So, = x45

Which gives us: 8 × = 9x i.e. x = min(s)

Hence, the duration of each period would be 40 minutes.

Chapter 11: Direct and Inverse Proportions

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Exercise 11.2

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Chapter 11: Direct and Inverse Proportions

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Glossary

Exercise 11.2