## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 9 Direct and Inverse Variation Objective Type Questions

Mental Maths
Question 1.
Fill in the blanks:
(i) Two quantities are said to be in direct variation if increase (or decrease) in one quantity causes in ………….. other quantity.
(ii) Two quantities x andy are said to be in inverse variation if xy is …………..
(iii) The total cost of articles varies ………….. to the number of articles purchased.
(iv) More work is done in ………….. time.
(v) The time taken to finish a work varies ………….. to the number of men at work.
(vi) The speed of a moving object varies inversely to the ………….. to cover a certain distance.
(vii) The number of articles varies ………….. with the cost per article, if a fixed amount is available.
(viii)Remuneration is in ………….. of work done.
Solution:
(i) Two quantities are said to be in direct variation if increase (or decrease)
in one quantity causes increase or decrease in other quantity.
(ii) Two quantities x andy are said to be in inverse variation if xy is constant.
(iii) The total cost of articles varies directly to the number of articles purchased.
(iv) More work is done in more time.
(v) The time taken to finish a work varies inversely to the number of men at work.
(vi) The speed of a moving object varies inversely
to the time taken to cover a certain distance.
(vii) The number of articles varies inversely with the cost per article,
if a fixed amount is available.
(viii) Remuneration is in a proportion of work done.

Question 2.
State whether the following statements are true (T) or false (F):
(i) Two quantities x andy are said to be in inverse variation if $$\frac{x}{y}$$ is constant.
(ii) Number of days needed to complete the work = $$\frac{1}{\text { one day’s work }}$$
(iii) Two quantities x andy are said to be in direct variation if x = ky, where k is constant of variation.
(iv) The work done varies inversely to the number of men at work.
(v) In the given time, the distance covered by a moving object varies directly to its speed.
(vi) If A can complete a work in n days, then A’s one day’s work is $$\frac{1}{n}$$ of the work, n
(vii) More the money deposited in a bank, more is the interest earned.
(viii) If the number of articles purchased increases the total cost decreases.
(ix) At the same time length of shadow is in direct variation with length of the object.
(x) The distance covered varies inversely to the consumption of petrol.
Solution:
(i) Two quantities x and y are said to be in inverse variation
if $$\frac{x}{y}$$ is constant. False
Correct:
Two quantities x and y are said to be in inverse variation if xy is constant.
(ii) Number of days needed to complete the work = $$\frac{1}{\text { one day’s work }}$$ True
(iii) Two quantities x andy are said to be in direct variation
if x = ky, where k is constant of variation. True
(iv) The work done varies inversely to the number of ipen at work. False
Correct:
The work done varies directly to the number of men at work.
(v) In the given time, the distance covered by a moving object
varies directly to its speed. True
(vi) If A can complete a work in n days,
then A’s one day’s work is $$\frac{1}{n}$$ of the work. True
(vii) More the money deposited in a bank, more is the interest earned. True
(viii)If the number of articles purchased increases the total cost decreases. False
Correct:
It more articles increases, more cost increase.
(ix) At the same time length of shadow is in the
direct variation with length of the object.
True
(x) The distance covered varies inversely to the consumption of petrol. False
Correct:
It varies directly not inversely.

Multiple Choice Questions
Choose the correct answer from the given four options (3 to 13):
Question 3.
Two quantities x and y are said to be in inverse variation if
(a) xy = k
(b) x ∝ $$\frac{1}{y}$$
(c) x = $$\frac{k}{y}$$
(d) all of these
Solution:
x and y vary inversely
∴ xy = constant or x ∝ $$\frac{1}{y}$$
x = $$\frac{\text { constant }}{y}$$ are correct
All of these. (d)

Question 4.
If 12 metre wire costs ₹24, then the cost of 8 metre wire is
(a) ₹16
(b) ₹20
(c) ₹12
(d) ₹18
Solution:
Cost of 12 metre wire = ₹24
Cost of 8 metre wire = x
∴ 12 : 8 :: 24 : x ⇒ x = $$\frac{8 \times 24}{12}$$ = 16
∴ Cost of 8 metre wire = ₹16

Question 5.
If 5 kg wheat cost ₹60, then cost of 20 kg wheat is
(a) ₹200
(b) ₹210
(c) ₹220
(d) ₹240
Solution:
Cost of 5 kg of wheat = ₹60
Then cost of 20 kg of wheat
5 : 20 :: 60 : x (more wheat, more cost)
$$\frac{5}{20}=\frac{60}{x} \Rightarrow x=\frac{20 \times 60}{5}$$ = ₹240(d)

Question 6.
If 10-men can complete a work in 6 days, then 30 men can complete the same work in
(a) 2 days
(b) 3 days
(c) 4 days
(d) 5 days
Solution:
10 men can complete a work in = 6 days
30 men will complete it in less days (say × day)
∴ $$\frac{30}{10}=\frac{6}{x} \Rightarrow x=\frac{6 \times 10}{30}$$ = 2 days (a)

Question 7.
A car travels 80 km in 5 litres of petrol, then the distance covered by it in 15 litres of petrol is
(a) 400 km
(b) 240 km
(c) 200 km
(d) 100 km
Solution:
In 5 litres, a car travels = 80 km
In 15’litres, the car will travel x km
∴ 5 : 15 :: 80 : x
⇒ x = $$\frac{15 \times 80}{5}$$ = 240
∴ 240 km distance (b)

Question 8.
In a mess, there was enough food for 200 students for 20 days. If 50 new students joined them, then the food will last for
(a) 15 days
(b) 16 days
(c) 17 days
(d) 18 days
Solution:
Food lasts for 200 students for = 20 days
Let first will last for 200 + 50 = 250 students = x days
More students, less days
200 : 250 :: 20 : x
By inverse variation,
250 : 200 :: 20 : x
x = $$\frac{200 \times 20}{250}$$ = 16 days (b)

Question 9.
3 persons can paint a house in 8 days, then 4 persons can paint the same house in
(a) 5 days
(b) 6 days
(c) 7 days
(d) none of these
Solution:
3 persons can paint a house in 8 days.
4 persons will paint it in less days.
∴ $$\frac{4}{3}=\frac{8}{x} \quad \Rightarrow x=\frac{3 \times 8}{4}$$= 6 days (b)

Question 10.
A photograph of bacteria is enlarged 100000 times attains a length of 5 cm, then actual length of the bacteria is
(a) 0.00005 cm
(b) 5 × 10-5
(c) 5 × 10-7
(d) all of these
Solution:
Photograph of bacteria is enlarged 100000 times attain a length of 5 cm.
∴ Actual length = $$\frac{5}{100000}$$ = 5.0 × 10-5 (d)
(∵ 0.00005 cm = 5.0 × 10-5 cm = 5 × 10-7 m)

Question 11.
A tree 12 metre high casts a shadow of length 8 metre. Height of the tree whose shadow is 6 metre in length is
(a) 6 m
(b) 9 m
(c) 15 m
(d) none of these
Solution:
Shadow of 12 m high tree = 8 m
Shadow of another tree = 6 m
Let is height = x m
∴ 8 : 6 :: 12 : x
∴ x = $$\frac{12 \times 6}{8}=\frac{72}{8}$$ = 9 m (b)

Question 12.
If 5 pipes can fill the tank in 1 hour, then 4 pipes will fill the tank in
(a) 75 minutes
(b) 70 minutes
(c) 65 minutes
(d) none of these
Solution:
5 pipes can fill a tank in 1 hour
Then 4 pipes will fill in more time
∴ $$\frac{4}{5}=\frac{1}{x} \Rightarrow x=\frac{5}{4}$$ hours = 75 minutes (a)

Question 13.
A tap fills a tank in 8 hours and another tap at the bottom empties it in 10 hours. If both work together, the tank will be filled in
(a) 18 hours
(b) 24 hours
(c) 36 hours
(d) 40 hours
Solution:
First tap’s 1 hour work for filling = $$\frac{1}{8}$$
Second tap’s 1 hour work for emptying = $$\frac{1}{10}$$
Both work for 1 hour = $$\frac{1}{8}-\frac{1}{10}=\frac{1}{40}$$
∴ The tank will be filled in 40 hours. (d)

Value-Based Questions
Question 1.
The cost of fuel for running a train is proportional to the speed generated in km/h. It costs ₹40 per hour when train is moving with 20 km/h. What would be the cost of fuel per hour, if the train is moving with 60 km/h?
Keeping the safety and fuel prices in mind, state the values promoted in the question.
Solution:
∵ Cost of fuel is proportional to the speed of the train.

∴ Cost of fuel = ₹120
Save the fuel, save the Nation.

Question 2.
A pipe can fill a tank in 9 hours. There is a leakage in the bottom of the tank due to which tank is filled in 12 hours. If the tank is full, how much time will leakage take to empty the tank? Should we repair the leakage tank? Should we repair the leakage of the tank immediately? What values are being promoted?
Solution:
A pipe can fill a tank in = 9 hours
Due to leakage at the bottom it is filled in = 12 hours
∴ Leakage empty the tank = $$\frac{1}{9}-\frac{1}{12}$$ = $$\frac{4-3}{36}=\frac{1}{36}$$
∴ The leakage will empty the fill tank in 36 hours.
Save the water, save the world.

Higher Order Thinking Skills (Hots)
Question 1.
If 8 labourers can earn ₹9000 in 15 days, how many labourers can earn ₹6300 in 7 days?
Solution:
8 labourers can earn ₹9000 in 15 dyas
To find : How many labourers will earn ₹6300 in 7 days
Less amount, less labourers
Less days, more labours
∴ Number of labourers = $$\frac{8 \times 6300 \times 15}{9000 \times 7}$$ = 12
OR
8 labourers earn in 15 days = ₹9000
120 labour days = ₹9000
1 labour day = $$\frac{9000}{120}$$ = ₹7.5
Hence, the number of labourers day required to earn ₹6300 in 7 days = $$\frac{6300}{75}$$ = 84 days
∴ The number of labourers required = $$\frac{84}{7}$$ = 12

Question 2.
Three typists working 8 hours a day type a document in 10 days. If only 2 typists are working, how many hours a day should they work to finish the job in 12 days?
Solution:
3 typist in 8 hours a day can type a document in = 10 days
2 typists in 12 days will finish the work by working x hour a day
Less typists, more hours a day
More days, less hours
∴ x = $$\frac{8 \times 3 \times 10}{2 \times 12}$$ = 10 hours a day
∴ They will work for 10 hours a day.