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## NCERT Solutions for Class 8 Maths Chapter 13 : Direct and Inverse Proportions Ex 13.1

## NCERT Solutions for Class 8 Maths Chapter 13 : Direct and Inverse Proportions Ex 13.2

**Question 1:**

**Which of the following are in inverse proportion?**

**(i)The no. 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 fixed journey and the speed of the vehicle.**

**(v)The population of a country and the area of land**

**per person.**

**Solution**:

(i) They are in inverse proportion because if there are more workers, then it will take lesser time to complete that job considering the ratio.

(ii) No, these are not in inverse proportion because if we take 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) They are in inverse proportion because with more speed, we may complete a

certain distance in a lesser time.

(v) They are in inverse proportion because if the population is increasing, then the area of the land per person will be decreasing accordingly.

**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 no. of winners?**

**Solution**:

A table of the given information is as follows

From the table, we obtain

1 × 100000 = 2 × 50000 = 100000

Thus, the no. of winners and the amount given to each winner are inversely

proportional to each other. ∴,

1 × 100000 = 4 ×x1

x1 = 25000

1 × 100000 = 5 ×x2

x2 = 20000

1 × 100000 = 8 ×x3

x3 = 12500

1 × 100000 = 10 ×x4

x4 = 10000

1 × 100000 = 20 ×x5

x5 = 5000

**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**

**(i)Are the no. of spokes and the angles formed between the**

**pairs of consecutive spokes in inversely 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 is40°?**

**Solution**:

A table of the given information is as follows.

From the given table, we obtain

4 × 90° = 360° = 6 × 60°

Thus, the no. of spokes and the angle between a pair of consecutive spokes are inversely

proportional to each other.

∴ 4 × 90° = x1 × 8

90*4/8=x1/1=45

Similarly, x2 =90*4/10= 36

And x3 =20

Thus, the following table is obtained.

(i) Yes, the no. 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.

∴ 4 × 90° = 15 × x

Thus, the angle between a pair of consecutive spokes of a wheel, which has 15 spokes in it, is 24°.

(iii) Let the no. of spokes in a wheel, which has 40º angles between a pair of consecutive spokes, be y

∴,4 × 90° = y × 40°

Thus, the no. 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 no. of the children is reduced by 4?**

**Solution**:

No. of remaining children = 24 − 4 = 20

Let the no. of sweets which each of the 20 students will get, be x. The

following table is obtained.

If the no. of students is lesser, then each student will get more no. of sweets thus

As we know this case is of inverse proportion

∴ 24 × 5 = 20 × x

x =6

Thus, each student 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?**

**Solution**:

Let the no. of days that the food will last if there were 10 more animals in the cattle be x. The following table is planned.

More the no. of animals, lesser will be the no. of days for which the food will last.

Thus, the no. of days the food will last and the no. of animals are inversely proportional to

each other.

∴,20 × 6 = 30 × x

x = 20 x 6/30 =4

Thus, the food will last for 4 days.

**Question 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?**

**Solution**:

Consider the no. of days required by 4 persons to complete the given job be x. Thus the following table is obtained:

If the no. of persons is more, then it will take lesser time to complete the job. Thus, the no.

of days and the no. of persons required to complete the job are inversely proportional to

each other.

∴ 4 × 3 = x × 4

x =4 x 3/4 = 3

Thus, the no. 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?**

**Solution**:

Let the no. of boxes filled, by using 20 bottles in each box, be x. The

following table is obtained.

More the no. of bottles, lesser will be the no. of boxes.

Thus, the no. of bottles and the no. of boxes required to pack these are inversely

proportional to each other.

∴, 12 × 25 = 20× x

x =15

Thus, the no. of boxes required to pack these bottles is 15.

**Question 8:**

**A factory required 42 machines to produce a given no. of articles in 63**

**days. How many machines would be required to produce the same no. of**

**articles in 54 days?**

**Solution**:

Consider the no. of machines required to produce articles in 54 days be x. The following table is obtained.

More the no. of machines, lesser will be the no. of days that it will take to produce the given no. of articles. Thus, this is a case of inverse proportion.

∴ 42 × 63 = 54× x

x=49

Thus, the required no. of machines to produce the given no. of articles in 54 days is 49.

**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?**

**Solution**:

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.

We know more is the speed of the car, less will be the time taken by it to reach the given destination. Thus, the speed and the time taken by the car are inversely proportional to each

other.

∴ 60 × 2 = 80× x

x =1.5

The time required by the car to reach the given destination is 1.5 hours.

**Question 10:**

**Two persons could fit new windows in 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?**

**Solution**:

(i)

Let the no. of days required by 1 man to fit all the windows be x.

The following table will be obtained.

We know that, less the no. of persons, more will be the no. of days required to fit all the windows. Thus, this is a case of inverse proportion. ∴,

∴ 2 × 3 = 1× x

x = 6

Thus, the no. of days taken by 1 man to fit all the windows is 6.

(ii)

Let the no. of persons required to fit all the windows in one day be y.

Lesser the no. of days, more will be the no. of persons required to fit all the windows. Thus, this is a case of inverse proportion. ∴,

∴ 2 × 3 = y × 1

y = 6

Thus, 6 persons 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 no.**

**of school hours to be the same?**

**Solution**:

Consider the duration of each period, when there are 9 periods a day in the school, be x minutes. The following table will be obtained

If there is more no. of periods a day in the school, then the duration of each period will be lesser. Thus, this is a case of inverse proportion. ∴

∴ 45 × 8 = x× 9

45 x8 / 9 = x

Thus, in this case, the duration of each period will be 40 minutes.