1. Which of the following are in inverse proportion?

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

Solution:

They are inversely proportional because if the number of workers increase the job will take less time to complete, also less workers will take more time to complete the job.

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

Solution:

They are not inversely proportional because more time require for more distance. Hence, time and distance covered in direct proportion.

(iii) Area of cultivated land and the crop harvested.

Solution:

They are not inversely proportional because more area of cultivated land will yield more crops. Hence, they are direct proportion.

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

Solution:

They are inversely proportional because it will take less time to complete the journey if the speed is increase.

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

Solution:

They are inversely proportional because If the population of a country increases, then the area of land per person will be decrease.

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	8	10	20
The prize for each winner (in ₹)	1,00,000	50,000	…….	…….	…….	……..	…..

Solution:

We observe that:

1 x 100,000 = 2 x 50,000

1,00,000 = 1,00,000

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

Now,

Let the blank spaces be a, b, c, d and e respectively.

2 x 20,000 = 4 x a

a = (2 x 50,000)/4

   = 25,000

2 x 50,000 = 5 x b

b = (2 x 50,000)/5

    = 20,000

2 x 50,000 = 8 x c

c = (2 x 50,000)/8

   = 12,500

2 x 50,000 = 10 x d

d = (2 x 50,000)/10

   = 10,000

2 x 50,000 = 20 x e

e = (2 x 50,000)/20

   = 5,000

Number of winners	1	2	4	5	8	10	20
The prize for each winner (in ₹)	1,00,000	50,000	25,000	20,000	12,500	10,000	5,000

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 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 degree?

Solution:

From the table we observe that the number of spokes are increasing and the angle between a pair of consecutive spokes is decreasing. So, it is inversely proportion and angle at the centre of a circle is 360^o.

4 x 90^o = 6 x 60^o

360^o = 360^o

Let the blank space can say a, b and c

4 x 90^o = 8 x a

a = 45^o

4 x 90^o = 10 x b

b = 36^o

4 x 90^o = 12 x c

c = 30^o

required table is:

Number of spokes	4	6	9	10	12
The angle between a pair of consecutive spokes	90o	60o	45o	36o	30o

Now,

(i) Yes, they are in inversely proportion.

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

4 x 90^o = 15 x X

            x = 24^o.

(iii) The number of spokes would be needed:

If the angle between two consecutive spokes is 40⁰, then

4 x 90^o = y x 40^o

Y = 9

Hence, the required number of spokes = 9

  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?

Solution:

According to question:

Number of children	Number of sweets
24	5
24 – 4 = 20	b

They are inversely proportional.

Let required number of sweets be b

24 x 5 = 20 x b

b = 24 x 5 /20

   =6

Hence, the required number of sweets = 6

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:

According to question:

Number of animals	Number of days
20	6
20 + 10 = 30	b

They are inversely proportional.

Let the required number of days be b.

Total number of animals = 20+10 = 30

Now,

20 x 6 = 30 x b

b = 4

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?

Solution:

According to question:

Number of persons	Number of days
3	4
4	b

Here, it will take less number of days to complete the job when the number of persons will increase. Hence, they are inversely proportional.

Let time taken to complete the job be “b” days.

Now,

3 x 4 = 4 x b

b = 3

Hence, 4 persons will take 3 days to complete the job.

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:

According to question:

Number of boxes	Number of bottles per boxes
25	12
b	20

They are inversely proportional.

Let the required number of boxes be b.

Now,

25 x 12 = b x 20

b = 15

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?

Solution:

According to question:

Number of machines	Number of days
42	63
b	54

They are inversely proportional.

Let the required number of machines be b.

Now,

42 x 63 = b x 54

b = 49

Hence, 49 machines would be required to produce the article.

9. A car takes 2 hours to reach a destination by travelling at the speed of 60 km/hr. How long will it take when the car travels at the speed of 80 km/hr?

Solution:

According to question:

Speed (km/h)	Time (h)
60	2
80	b

They are inversely proportional.

Let the required number of hours be b.

Now,

60 x 2 = 80 x b

b = (60 x 2)/80

    = 3/2

    = 1.5 h

Hence, required time = 1.5 hours

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?

Solution:

According to question:

Number of persons	Number of days
2	3
(i)1 = (2-1)	b
(ii)p	1

(i)From the table, we observe that they are inversely proportional.

 Let the required number of days be b.

2/1 = b/3

b = 6

Hence, the required number of days = 6

(ii)From the table, we observe that they are inversely proportional.

Let the required number of persons be p.

2/p = 1/3

p = 6

Hence, the required number of persons = 6

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?

Solution:

According to question:

Number of periods	Periods duration (minutes)
8	45
9	b

They are inversely proportional.

Let the required duration of each period be b.

8/9 = b/45

8×45 = 9b

b = 40

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