CLASS 8TH CHAPTER- 13 DIRECT AND INVERSE PROPRATION NCERT SOLUTION

Exercise 13.1 | Class 8th Mathematics

1. Following are the car parking charges near a railway station up to.
   4 hours – ₹ 60
   8 hours – ₹ 100
   12 hours – ₹ 140
   24 hours – ₹ 180
   Check if the parking charges are in direct proportions to the parking time.
   
   Solution:
   We have the ratio of time period and the parking charge.
   
   Hence the given quantities are not directly proportional.

2. A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. In the following table, find the parts of base that need to be added.

Solution:
Let the number to be filled in the blanks be a, b, c and d respectively.

3. 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?
Solution:
Let the required red pigment be x part.

Hence, the required amount of red pigment = 24 parts.

4. A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?
Solution:
Let the required number of bottles be x.

Hence the required number of bottles = 700.

5. 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?

Solution:
Let the actual length be x cm.

6. 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?

Solution:
Let the required length of the model ship be x m.

7. 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?
Solution:
Let x be the number of sugar crystals needed.

8. Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road of 72 km. What would be her distance covered in the map?
Solution:
Let the required distance be x km.

Hence the distance covered in the map = 4 cm.

9. 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 5 m long.
Solution:
(i) Let the required length of shadow be x m.

10. A loaded truck travels 14 km in 25 minutes. If the speed remains the same, how far can it travel in 5 hours?
Solution:
Let the required distance be x km.

Hence the required distance = 168 km.

Exercise 13.2 | Class 8th Mathematics

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 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) As the number of workers increase, the job will take less time to complete. Hence, they are inversely proportional.
   (ii) For more time, more distance to travel. Hence, they are not inversely proportional.
   (iii) More area of land cultivated, more crop to harvest. Hence, they are not inversely proportional.
   (iv) If speed is increased, it will take less time to complete the fixed journey. Hence, they are inversely proportional.
   (v) If the population of a country increases, then the area of land per person will be decreased. Hence, they are 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	1	2	4	5	8	10	20
The prize for each winner (in ₹)	1,00,000	50,000	–	–	–	–	–

Solution:
Let, the blank spaces be denoted by a, b, c, d and e.
So, we observe that 1 × 100,000 = 2 × 50,000
⇒ 1,00,000 = 1,00,000
Hence they are inversely proportional.
2 × 50,000 = 4 × a

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.

Number of spokes	4	6	8	10	12
The angle between a pair of consecutive spokes	90°	60°	–	–	–

(i) Are the number of spokes and the angle 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°?
Solution:
From the above table, we observe that
4 × 90° = 6 × 60°
360° = 360°
Thus the two quantities are inversely proportional.
Let the blank spaces be denoted by a, b, and c.
4 × 90° = 8 × a

Hence, the required table is

Number of spokes	4	6	8	10	12
The angle between a pair of consecutive spokes	90°	60°	45°	36°	30°

(i) Yes, they are in inverse proportion
(ii) If the number of spokes is 15, then
4 × 90° = 15 × x
x = 4×9015 = 24°
(iii) If the angle between two consecutive spokes is 40°, then
4 × 90° = y × 40°
y = 4×9040 = 9 spokes.
Thus 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 decreased by 4?
Solution:

Number of children	Number of Sweets
24	5
(24 – 4) or 20	a

We observe that on increasing the number of children, number of sweets got by each will be less. So, they are in inverse proportion.
x₁y₁ = x₂y₂
where x₁ = 24, y₁ = 5, x₂ = 20
and y₂ = a(let)
24 × 5 = 20 × a
a = 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:
If the number of animals increases, then it will take fewer days to last.
Then the two quantities are in inverse proportions.

Number of animals	Number of days
20	6
(20 + 10) or 30	P

Let the required number of days be p.
x₁y₁ = x₂y₂
where x₁ = 20, y₁ = 6, x₂ = 3
and y₂ = p (let)
20 × 6 = 30 × p
p = 4
Hence the required number of days = 4.

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:
If the number of persons is increased, it will take less number of days to complete the job.
Thus, the two quantities are in inverse proportion.

Number of persons	Number of days
3	4
4	k

Let the required number of days be k.
x₁y₁ = x₂y₂
3 × 4 = 4 × k
k = 3 days.
Hence, the required number of days = 3.

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:
If the number of bottles is increased then the required number of boxes will be decreased. Thus the two quantities are in inverse proportion.

Number of boxes	Number of bottles per box
25	12
x	20

Let the required number of boxes be x.
x₁y₁ = x₂y₂
25 × 12 = x × 20
x = 15
Hence, the required number of boxes = 15.

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:
If the number of machines is increased then less number of days would be required to produced the same number of articles.
Thus, the two quantities are in inverse proportion.

Number of machines	Number of days
42	63
x	54

Let the required number of machines be x.
x₁y₁ = x₂y₂
42 × 63 = x × 54
x = 49
Hence, the required number of machines is 49.

9. A car takes 2 hours to reach a destination by traveling at a speed of 60 km/h. How long will it take when the car travels at the speed of 80 km/h?
Solution:
On increasing the speed, it will take less time to travel a distance.
Thus the two quantities are in inverse proportions.

Speed in km/h	Time in hour
60	2
80	x

Let the required times be x hours.
x₁y₁ = x₂y₂
60 × 2 = 80 × x
x = 32 hours = 112 hrs.
Hence, the required time = 112 hours.

10. Two persons could fit new windows in a house in 3 days.
(i) One of the people 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:
On increasing the number of persons, less time will be required to complete a job.
Thus, the quantities are in inverse proportion.

Number of persons	Number of days
2	3
(i) 1(2 – 1)	x
(ii) y	1

(i) Let the required number of days be x.
x₁y₁ = x₂y₂
2 × 3 = 1 × x
x = 6
Hence, the required number of days = 6
(ii) Let the required number of persons be y.
x₁y₁ = x₂y₂
2 × 3 = y × 1
y = 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:
On increasing the duration of periods, the number of periods will be reduced.
Thus, the two quantities are in inverse proportion.

Number of periods	Duration of periods in minutes
8	45
9	x

Let the required duration of each period be x.
x₁y₁ = x₂y₂
8 × 45 = 9 × x
x = 40 minutes
Hence, the required duration of period = 40 minutes.

Extra Questions | Class 8th Mathematics

Direct and Inverse Proportions Class 8 Extra Questions Very Short Answer Type

1. A train is moving at a uniform speed of 100 km/h. How far will it travel in 20 minutes?
Solution:
Let the distance travelled by train in 20 minutes be x km.

Since the speed is uniform, the distance travelled will be directly proportional to time.

Hence, the required distance is 3313 km.

2. Complete the table if x and y vary directly.

Solution:
Let the blank spaces be filled with a, b and c.


Hence, the required values are a = 7, b = 15 and c = 7.5.

3. If the cost of 20 books is ₹ 180, how much will 15 books cost?
Solution:
Let the required cost be ₹ x.
Here, the two quantities vary directly.

4. If x₁ = 5, y₁ = 7.5, x₂ = 7.5 then find y₂ if x and y vary directly.
Solution:
Since x and y vary directly.

Hence, the required value is 11.25.

5. If 3 kg of sugar contains 9 × 10⁸ crystals. How many sugar crystals are there in 4 kg of sugar?
Solution:
Let the required number of crystals be x.

Hence, the required number of crystals = 1.2 × 10⁹.

6. If 15 men can do a work in 12 days, how many men will do the same work in 6 days?
Solution:
Let the required number of men be x.
Less days → more men.
Thus the two quantities are inversely proportional to each other.

Hence, the required number of days = 30.

7. A train travels 112 km in 1 hour 30 minutes with a certain speed. How many kilometres it will travel in 4 hours 45 minutes with the same speed?
Solution:
Let the required distance be x km.
More distance → more time
Thus, the two quantities are directly proportional.


Hence, the required distance = 354.6 km.

8. The scale of a map is given as 1 : 50,000. Two villages are 5 cm apart on the map. Find the actual distance between them.
Solution:
Let the map distance be x cm and actual distance be y.
1 : 50,000 = x : y

Hence, the required distance = 250 km.

9. 8 pipes are required to fill a tank in 1 hr 20 min. How long will it take if only 6 pipes of the same type are used?
Solution:
Let the required time be ‘t’ hours.

10. 15 men can build a wall in 42 hours, how many workers will be required for the same work in 30 hours?
Solution:
Let the required number of workers be x.
The number of workers, faster will they do the work.
So, the two quantities are inversely proportional.

x₁y₁ = x₂y₂
⇒ 42 × 15 = 30 × x
⇒ x = 21
Hence, the required number of men = 21

Direct and Inverse Proportions Class 8 Extra Questions Short Answer Type

11. The volume of a gas V varies inversely as the pressure P for a given mass of the gas. Fill in the blank spaces in the following table:

Solution:
Since volume and pressure are inversely proportional.
PV = K

12. The cost of 5 metres of cloth is ₹ 210. Tabulate the cost of 2, 4, 10 and 13 metres of cloth of the same type.
Solution:
Let the length of the cloth be x m and its cost be ₹ y. We have the following table.

13. Six pumps working together empty a tank in 28 minutes. How long will it take to empty the tank if 4 such pumps are working together?
Solution:
Let the required time be t minutes.

Less pump → More time
Since there is an inverse variation.
x₁y₁ = x₂y₂
6 × 28 = 4 × x
x = 42
Hence, the required time = 42 minutes.

14. Mohit deposited a sum of ₹ 12000 in a Bank at a certain rate of interest for 2 years and earns an interest of ₹ 900. How much interest would be earned for a deposit of ₹ 15000 for the same period and at the same rate of interest?
Solution:
Let the required amount of interest be ₹ x.


Hence, the required amount of interest = ₹ 1125.

15. A garrison of 120 men has provisions for 30 days. At the end of 5 days, 5 more men joined them. How many days can they sustain on the remaining provision?
Solution:
Let the number of days be x.

[∵ Remaining days = 30 – 5 = 25]
[Total men = 120 + 5 = 125]
Since there is an inverse variation.
x₁y₁ = x₂y₂
120 × 25 = 125 × x
x = 24
Hence, the required number of days be 24.

16. In a scout camp, there is food provision for 300 cadets for 42 days. If 50 more persons join the camp, for how many days will the provision last? (NCERT Exemplar)
Solution:
More the persons, the sooner would be the provision exhausted. So, this is a case of inverse proportion.
Let the required number of days be x.
Hence, 300 × 42 = (300 + 50) × x
300 × 42 = 350 × x
x = 36

17. If two cardboard boxes occupy 500 cubic centimetres space, then how much space is required to keep 200 such boxes? (NCERT Exemplar)
Solution:
As the number of boxes increases, the space required to keep them also increases.
So, this is a case of direct proportion.

Thus, the required space is 50,000 cubic centimetres.

18. Under the condition that the temperature remains constant, the volume of gas is inversely proportional to its pressure. If the volume of gas is 630 cubic centimetres at a pressure of 360 mm of mercury, then what will be the pressure of the gas if its volume is 720 cubic centimetres at the same temperature? (NCERT Exemplar)
Solution:
Given that, at constant temperature pressure and volume of a gas are inversely proportional.
Let the required pressure be x.

Therefore, the required pressure is 315 mm of mercury.