NCERT Exemplar Problems Class 8 Mathematics Chapter 10 Direct and Inverse Proportion

Question. 1 Both u and v vary directly with each other. When u is 10, v is 15, which of the following is not a possible pair of corresponding values of u and v?
(a)2 and 3 (b) 8 and 12 (c) 15 and 20 (d) 25 and 37.5
Solution.

Question. 2 Both x and y vary inversely with each other. When x is 10, y is 6, which of the following is not a possible pair of corresponding values of x and y?
(a) 12 and 5 (b) 15 and 4 (c) 25 and 2.4 (d) 45 and 1.3
Solution.

Question. 3 Assuming land to be uniformly fertile, the area of land and the yield on it vary
(a) directly with each other
(b) inversely with each other
(c) neither directly nor inversely with each other
(d) sometimes directly and sometimes inversely with each other
Solution. (a) If land to be uniformly fertile, then the area of land and the yield on it vary directly with each other.
Hence, option (a) is correct.
Note Two quantities x and y are said to be in direct proportion, if they increase or decrease together in such a manner that the ratio of their corresponding values remains constant.

Question. 4 The number of teeth and the age of a person vary
(a) directly with each other
(b) inversely with each other
(c) neither directly nor inversely with each other
(d) sometimes directly,and sometimes inversely with each other
Solution. (d) The number of teeth and the age of a person vary sometimes directly and sometimes inversely with each other, we cannot predict about the number of teeth with exactly the age of a person. It change with person-to-person.
Hence, option (d) is correct.




Question. 5 A truck needs 54 litres of diesel for covering a distance of 297 km. The diesel required by the truck to cover a distance of 550 km is (a) 100 litres (b) 50 litres (c) 25.16 litres (d) 25 litres
Solution.

Question. 6 By travelling at a speed of 48 km/h, a car can finish a certain journey in 10 hours. To cover the same distance in 8 hours, the speed of the car should be
(a) 60 km/h (b) 80 km/h (c) 30 km/h (d) 40 km/h
Solution.

Question. 7 In which of the following cases, do the quantities vary directly with each other?

Solution. (a) In option (a),
x = 0.5,2, 8, 32 and y = 2, 8, 32,128
If we multiply x with 4, we get the directly required result as same as shown in corresponding y. In this case, as the value of x increases, the value of y also increases. Hence, option (a) is correct.

Question. 8 Which quantities in the previous question vary inversely with each other?
(a) x and y (b) p and q (c) r and s (d) u and v
Solution.

Question. 9 Which of the following vary inversely with each other?
(a) Speed and distance covered (b) Distance covered and taxi fare
(c) Distance travelled and time taken (d) Speed and time taken
Solution. (d) We know that, when we increases the speed, then the time taken by vehicle decreases.
Hence, speed and time taken vary inversely with each other.
So, option (d) is correct.

Question. 10 Both x and y are in direct proportion, then and are
(a) in indirect proportion
(b) in inverse proportion
(c) neither in direct nor in inverse proportion
(d) sometimes in direct and sometimes in inverse proportion
Solution. (b) If both x arid y are in directly proportion, then and are in inverse proportion.
Hence, option (b) is correct.
Note Two quantities x and y are said to be in inverse proportion, if an increase in x cause a proportional decrease in y and vice-versa.

Question. 11 Meenakshee cycles to her school at an average speed of 12 km/h and takes 20 minutes to reach her school. If she wants to reach her school in 12 minutes, her average speed should be (a) 20/3 km/h (b) 16 km/h (c) 20 km/h. (d) 15 km/h
Solution.

Question. 12 100 persons had food provision for 24 days. If 20 persons left the place, the provision will last for
(a) 30 days (b) 96/5 days (c) 120 days (d) 40 days
Solution.





Question. 13 If two quantities x and y vary directly with each other, then
(a) remains constant (b) x – y remains constant
(c) x + y remains constant ‘ (d)ix y remains constant
Solution. (a) If two quantities x and y vary directly with each other, then = k = constant.
Since, in direct proportion, both x and y increases or decreases together such a manner that the ratio of their corresponding value remains constant. Hence, option (a) is correct.

Question. 14 If two quantities p and q vary inversely with each other, then
(a) remains constant (b) p + q remains constant (c) p x q remains constant (d) p – q remains constant
Solution. (c) If two quantities p and q vary inversely with each other, then p x q remains constant.
Since, in inverse proportion, an increase in p cause a proportional decrease in q and vice-versa.
Hence, option (c) is correct.

Question. 15 If the distance travelled by a rickshaw in one hour is 10 km, then the distance travelled by the same rickshaw with the same speed in one minute is
(a)m (b)m (c)1000m (d)m
Solution.

Question. 16 Both x and y vary directly with each other and when x is 10, y is 14,
which of the following is not a possible pair of corresponding values of x and y?
(a) 25 and 35 (b) 35 and 25
(c) 35 and 49 (d) 15 and 21
Solution.

Fill in the Blanks
In questions 17 to 42, fill in the blanks to make the statements true.
Question. 17 If x = 5y, then x and y vary——— with each other.
Solution.

Question. 18 If xy = 10, then x and y vary————with each other.
Solution.

Question. 19 When two quantities x and y are in——-proportion or vary——-they are written as
Solution.When two quantities x and y are in direct proportion or vary directly, they are written as [see definition of direct proportion]

Question. 20 When two quantities x and y are in——-proportion or vary———-they are written as
Solution. When two quantities x and y are in inverse proportion or vary inversely, they are written as [see definition of inverse proportion]

Question. 21 Both x and y are said to vary——–with each other, if for some positive number k, xy =k.
Solution. Both x and y are said to vary inversely with each other, if for some positive number k,xy = k. [see condition of inverse proportion]

Question. 22 x and y are said to vary directly with each other, if for, some positive number k,———-= k.
Solution. x and y are said to vary directly wifh’ether, if for some positive number k, =k.

Question. 23 Two quantities are said to vary——— with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.
Solution. Two quantities are said to vary directly with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.

Question. 24 Two quantities are said to vary——–with each other, if an increase in one causes a decrease in the other in such a manner that the product of their corresponding values remains constant.
Solution. Two quantities are said to vary inversely with each other, if increase in one cause a decrease in the other in such a manner that the product of their corresponding values remains constant.

Question. 25 If 12 pumps can empty a reservoir in 20 hours, then time required by 45 such pumps to empty the same reservoir in—-hours.
Solution.

Question. 26 If x varies inversely as y then

Solution.

Question. 27 If x varies directly as y, then

Solution.

Question. 28 When the speed remains constant, the distance travelled is——–proportional to the time.
Solution. When the speed remains constant, the distance travelled is directly proportional to the time.
e.g. If 10 km cover in 10 min with uniform speed, then 20 km cover in 20 min with same speed.

Question. 29 On increasing a, b increases in such a manner that remains——and positive, then a and b are said to vary directly with each other.
Solution. On increasing a, b increases in such a manner that remains constant and positive, then a and b are said to vary directly with each other.

Question. 30 If on increasing a, b decreases in such a manner that—– remains——–and positive, then a and b are said to vary inversely with each other.
Solution. If on increasing a, b decreases in such a manner that ab remains constant and positive, then a and b are said to vary inversely with each other. [see definition of inverse proportion]

Question. 31 If two quantities x and y vary directly with each other, then——— of their corresponding values remains constant.
Solution. If two quantities x and y vary directly with each other, then ratio of their corresponding values remains constant. [see definition of direct proportion]

Question. 32 If two quantities p and q vary inversely with each other, then————- of their corresponding values remains constant.
Solution. If two quantities p and q vary inversely with each other, then product of their corresponding values remains constant.

Question. 33 The perimeter of a circle and its diameter vary—–with each other.
Solution.

Question. 34 A car is travelling 48 km in one hour. The distance travelled by the car in 12 minutes is ———.
Solution.

Question. 35 An auto rickshaw takes 3 hours to cover a distance of 36 km. If its speed is increased by 4 km/h, the time taken by it to cover the same distance is———–.
Solution.

Question. 36 If the thickness of a pile of 12 cardboard sheets is 45 mm, then the thickness of a pile of 240 sheets is——cm.
Solution.

Question. 37 If x varies inversely as y and x = 4 when y = 6, when x = 3, then value of y is—.
Solution.

Question. 38 In direct proportion —————-
Solution.

Question. 39 In case of inverse proportion,

Solution.

Question. 40 If the area occupied by 15 postal stamps is 60 , then the area occupied by 120 such postal stamps will be———.
Solution.

Question. 41 If 45 persons can complete a work in 20 days, then the time taken by 75 persons will be—–hours.
Solution.

Question. 42 Devangi travels 50 m distance in 75 steps, then the distance travelled in 375 steps is——– km.
Solution.

True/False
In questions from 43 to 59, state whether the statements are True or False.
Question. 43 Two quantities x and y are said to vary directly with each other, if for some rational number k, xy =k.
Solution. False
Two quantities x and y are said to vary directly with each other, if = k (constant)

Question. 44 When the speed is kept fixed, time and distance vary inversely with each other.
Solution. False
When the speed is kept fixed, time and distance vary directly with each other.

Question. 45 When the distance is kept fixed, speed and time vary directly with each other.
Solution. False
When the distance is kept fixed, speed and time vary indirectly/inversely with each other. Since, if we increase speed, then taken time will less and vice-versa.

Question. 46 Length of a side of a square and its area vary directly with each other.
Solution. False
Length of a side of a square and its area does not vary directly with each other, e.g. Let a be length of each side of a square.
So, area of the square = =
So, if we increase the length of the side of a square, then their area increases but not directly.

Question. 47 Length of a side of an equilateral triangle and its perimeter vary inversely with each other.
Solution. False
Length of a side of an equilateral triangle and its perimeter vary directly with each other, e.g. Let a be the side of an equilateral triangle. So, perimeter = 3 x (Side) = 3 x a = 3a . So, if we increase the length of side of the equilateral triangle, then their perimeter will also increases.

Question. 48 If d varies directly as , then we can write d = k, where k is some constant.
Solution. False
If d varies inversely as , then we can write d = k, where k is some constant.
Since, two quantities x and y are said to be in Inverse proportion, if an increases in x cause a proportional decreases in y and vice-versa, in such a manner that the product of their corresponding values remains constant.

Question. 49 If a tree 24 m high casts a shadow of 15 m, then the height of a pole that casts a shadow of 6 m under similar conditions is 9.6 m.
Solution.

Question. 50 If x and y are in direct proportion, then (x -1) and (y -1) are also in direct proportion.
Solution.

Question. 51 If x and y are in inverse proportion, then (x +1) and (y +1) are also in inverse proportion.
Solution. False
If x and y are in inverse proportion, then xy = k (constant) e.g. Let x= 2 and y = 3
.-. xy = 2 x 3= 6. Now, x + 1=2 + 1 = 3 and y+ 1 = 3 + 1 = 4
Then, (x + 1)(y+1) = 3 x 4 = 12 [not in inverse proportion]
Hence, (x+ 1)and (y + 1) cannot be in inverse proportion.

Question. 52 If p and q are in inverse proportion, i.e. pq = k (constant), then (p + 2)and (q – 2) are also in inverse proportion.
Solution. False
If p and q are in inverse proportion, then
xy = k (constant)
e.g. Let p = 3andq = 4
Then, pq = 3×4 = 12
Now, p+ 2 = 3+ 2 = 5 and q-2 = 4-2 =2
(p + 2) (q – 2) = 5 x 2 = 10 [not in inverse proportion]
Henc, (p+2) and (q -2)cannot be in inverse proportion.

Question. 53 If one angle of a, triangle is kept fixed, then the measure of the remaining two angles vary inversely with each other.
Solution.

Question. 54 When two quantities are related in such a manner that, if one increases, the other also increases, then they always vary directly.
Solution. True
When two quantities are related in such a manner that if, one increases the other also increases, then they always vary directly.
Above statement is correct for direct proportion. It is a basic properties of direct proportion.

Question. 55 When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely.
Solution. True
When, two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely. Above statement is correct for inverse proportion. It is a basic properties of inverse proportion.

Question. 56 If x varies inversely as y and when x = 6, y = 8, then for x = 8, the value of y is 10.
Solution.

Question. 57 The number of workers and the time to complete a job is a case of direct proportion.
Solution. False
The number of workers and the time to complete a job is a case of indirect proportion, e.g. If 60 workers can complete a work in 10 days.
Then, 120 workers can complete the same work in 5 days.

Question. 58 For fixed time period and rate of interest, the simple interest is directly proportional to the principal.
Solution.

Question. 59 The area of cultivated land and the crop harvested is a case of direct proportion.
Solution. True
The area of cultivated land and the crop harvested is a case of direct proportion.
Since, the quantities of crop harvested is depend upon area of cultivated land.

In questions 60 to 62, which of the following vary directly and which vary inversely with each other and which are neither of the two ?
Question. 60 (i)The time taken by a train to cover a fixed distance and the speed of the train.
(ii) The distance travelled by CNG bus and the amount of CNG used.
(iii) The number of people working and the time to complete a given work.
(iv) Income tax and the income.
(v) Distance travelled by an auto-rickshaw and time taken.
Solution. (i) The time taken by a train to cover a fixed distance and the speed of the train are inversely proportional. ‘
e.g. Let a train cover 100 km in 1 h with speed 100 km/h.
Then, the same train cover 100 km in 30 min with speed 200 km/h.
(ii) The distance travelled by CNG bus and the amount of CNG used are directly proportional.
e.g. Let a CNG bus can travelled 10 km in 1 kg of CNG.
Then, the same CNG bus travelled 20 km in 2 x 1 = 2 kg of CNG.
(iii) The number of people working and the time to complete a given work are inversely proportional to each other.
e.g. Let 20 workers can complete a work in 1day.
Then, 10 workers can complete the same work in 2 days.
(iv) Income tax and the income are directly proportional to each other, e.g. Let Mr X have 4.5 lakh annual income.
Then, he pay 10% income tax on his income.
But if Mr X have 5.5 lakh annual income, then he has to pay 30% income tax on his salary/income.
(v) Distance travelled by an auto rickshaw and time taken are directly proportional to each other.
e.g. Let an auto rickshaw takes 2 h to travel 10 km.
Then, it will take 4 h to travel 20 km.

Question. 61 (i) Number of students in a hostel and consumption of food.
(ii) Area of the walls of a room and the cost of white washing the walls.
(iii) The number of people working and the quantity of work.
(iv) Simple interest on a given sum and the rate of interest.
(v) Compound interest on a given sum and the sum invested.
Solution.

Question. 62 (i) The quantity of rice and its cost.
(ii) The height of a tree and the number of years.
(iii) Increase in cost and number of shirts that can be purchased, if the budget remains the same.
(iv) Area of land and its cost.
(v) Sales tax and the amount of the bill.
Solution.(i) The quantity of rice and its cost are directly proportional to each other, e.g. Let 1 kg of rice price = Rs 40
Then, 2 kg of rice price = Rs 2 x 40 = Rs 80
(ii) The height of a tree and the number of years are neither directly nor inversely proportional to each other.
(iii) Increase in cost and number of shirts that can be purchased, if the budget remains the same are inversely proportional to each other, e.g. Let 2 shirts price = Rs 800 After increasing in price of each shirt,
1 shirt price became Rs 800
where budget = Rs 800

Question. 63 If x varies inversely as y and x = 20 when y = 600, find y when x = 400.
Solution.

Question. 64 The variable x varies directly as y and x – 80 when y is 160. What is y when x is 64?
Solution.





Question. 65 l varies directly as m and l = 5, when m = . Find l when m = .
Solution.

Question. 66 If x varies inversely as y and y =60 when x = 1.5. Find x, when y = 4.5.
Solution.

Question. 67 In a camp, there is enough flour for 300 persons for 42 days. How long will the, flour last, if 20 more persons join the camp?
Solution.

Question. 68 A contractor undertook a contract to complete a part of a stadium in 9 months with a team of 560 persons. Later on, it was required to complete the job in 5 months. How many extra persons should he employ to complete the work?
Solution.

Question. 69 Sobi types 108 words in 6 minutes. How many words would she type in half an hour?
Solution.

Question. 70 A car covers a distance in 40 minutes with an average speed of 60 km/h. What should be the average speed to cover the same distance in 25 minutes?
Solution.

Question. 71 It is.given that l varies directly as m.
(a) Write an equation which relates l and m.
(b) Find the constant of proportion (k), when l is 6, then m is 18.
(c) Find l, when m is 33.
(d) Find m, when l is 8.
Solution.

Question. 72 If a deposit of Rs 2000 earns an interest of Rs 500 in 3 years, how much interest would a deposit of Rs 36000 earn in 3 years with the same rate of simple interest?
Solution.

Question. 73 The mass of an aluminium rod varies directly with its length. If a 16 cm long rod has a mass of 192 g, find the length of the rod whose mass is 105 g.
Solution.

Question. 74 Find the values of x and y, if a and b are in inverse proportion.
(a) 12 x 8 (b) 305 y
Solution.

Question. 75 If Naresh walks 250 steps to cover a distance of 200 metres, find the distance travelled in 350 steps.
Solution.

Question. 76 A car travels a distance of 225 km in 25 litres of petrol. How many litres of petrol will be required to cover a distance of 540 kilometres by this car?
Solution.

Question. 77 From the following table, determine if x and y are in direct proportion or not.

Solution.

Question. 78 If a and b vary inversely to each other, then find the values of p, q, r; x, y, z and l, m, n.

Solution.

Question. 79 If 25 metres of cloth costs Rs 337.50, then
(a) what will be the cost of 40 metres of the same type of cloth? (b) what will be the length of the cloth bought for Rs 810?
Solution.

Question. 80 A swimming pool.can be filled in 4 hours by 8 pumps of the same type. How many such pumps are required, if the pool is to be filled in 2 hours?
Solution.

Question. 81 The cost of 27 kg of iron is ? 1080, what will be the cost of 120 kg of iron of the same quality?
Solution.

Question. 82 At a particular time, the length of the shadow of Qutub Minar whose height is 72 m is 80 m. What will be the height of an electric pole, the length of whose shadow at the same time is 1000 cm?
Solution.

Question. 83 In a hostel of 50 girls, there are food provisions for 40 days. If 30 more girls join the hostel, how long will these provisions last?
Solution.

Question. 84 Campus and Welfare Committee of school is planning to develop a blue shade for painting the entire school building. For this purpose, various shades are tried by mixing containers of blue paint and white paint. In each of the following mixtures, decide which is a lighter shade of blue and also find the lightest blue shade among all of them.

If one container has one litre paint and the building requires 105 litres for painting, how many container of each type is required to paint the building by lightest blue shade?
Solution.

Question. 85 Posing a Question Work with a partner to write at least five ratio statement about this quilt, which has white, blue and purple squares.

How many squares of each colour will be there in 12 such quilts?
Solution.

Question. 86 A packet of sweets was distributed among 10 children and each of them received 4 sweets. If it is distributed among 8 children, how many sweets will each child get?
Solution. The total number of children = 10
If each children received 4 sweets, then The total number of sweets = 10 x 4 = 40 sweets
If 40 sweets distributed between 8 children, then each get 40/8 i.e. 5 sweets.

Question. 87 44 cows can graze a field in 9 days. How many less/more cows will graze the same field in 12 days?
Solution.

Question. 88 30 persons can reap a field in 17 days. How many more persons should be engaged to reap the same field in 10 days?
Solution.

Question. 89 Shabnam takes 20 minutes to reach her school, if she goes at a speed of 6 km/h. If she wants to reach school in 24 minutes, what should be her speed?
Solution.

Question. 90 Ravi starts for his school at 8 : 20 am on his bicycle. If he travels at a speed of 10 km/h, then he reaches his school late by 8 minutes but on travelling at 16 km/h, he reaches the school 10 minutes early. At what time does the school start?
Solution.

Question. 91 Match each of the entries in Column I with the appropriate entry in Column II.

Solution.

Question. 92 There are 20 grams of protein in 75 grams of sauted fish. How many grams of protein is in 225 grams of that fish?
Solution.

Question. 93 Ms Anita has to drive from Jhareda to Ganwari. She measures a distance of 3.5 cm between these village on the map. What is the actual distance between the villages, if the map scale is 1 cm = 10 km?
Solution. The distance between Jhareda to Ganwari in the map = 3.5 cm Given scale, 1 cm = 10 km
So, actual distance between the villages = 35 x 10 = 35 km

Question. 94 A water tank casts a shadow 21 m long. A tree of height 9.5 m casts a shadow 8 m long at the same time. The length of the shadows are directly proportional to their heights. Find the height of the tank.

Solution.

Question. 95 The table shows the time four elevators take to travel various distances. Find, which elevator is fastest and which is slowest.

How much distance will be travelled elevators B and C separately in 140 sec? Who travelled more and by how much?
Solution.

Question. 96 A volleyball court is in a rectangular shape and its dimensions are directly proportional to the dimensions of the swimming pool given below. Find the width of the pool.

Solution.

Question. 97 A recipe for a particular type of muffins requires 1 cup of milk and 1.5 cups of chocolates. Riya has 7.5 cups of chocolates.If she is using the recipe as a guide, how many cups.of milk will she need to prepare muffins?

Solution.

Question. 98 Pattern B consists of four tiles like pattern A. Write a proportion involving blue dots and total dots in patterns A and B. Are they in direct proportion? If yes, write the constant of proportion.

Solution.

Question. 99 A Fowler throws a cricket ball at a speed of 120 km/h. How long does this ball take to travel distance of 20 m to each the batsman?
Solution.

Question. 100 The variable x is inversely proportional to y. If x increases by p%, then by what per cent will y decwsose?
Solution. The variable x is inversely proportional to y. xy = k (constant)
Since, we know that two quantities x and y are said to be in inverse proportion, if an increase in * cause a proportional decrease in y and vice-versa.So, we can say y decrease by p%.

Question. 101 Here is a keyboard of a harmonium.
(a) Find the ratio of white keys to black keys on the keyboard.

(b) What is the ratio of black keys to all keys on the given keyboard?
(c) This pattern of keys is repeated on larger keyboard. How many black keys would you expect to find on a keyboard with 14 such patterns?
Solution.

Question. 102 The following table shows the distance travelled by one of the new eco-friendly energy-efficient car travelled on gas.

Which type of properties are indicated by the table? How much distance will be covered by the car in 8 litres of gas?
Solution.On the basis of given table, the distance travelled by one of the new eco-friendly energy-efficient earns travelled on gas.
The car travelled 15 km In 1 L of gas.
The car travelled 7.5 km in 0.5 L of gas.
The car travelled 30 km in 2 L of gas.
This rate shows direct proportion between litres of gas and the distance cover.
The car can cover the distance in 8 L of gas = 8 x 15 = 120 km

Question. 103 Kritika is following this recipe for bread. She realises her sister used most of sugar syrup for her breakfast. Kritika has only cup of syrup,so she decides to make a small size of bread. How much of each ingredient shall she use?
Bread recipe
1 cup quick cooking oats 2 cups bread flour
cup sugar syrup 1 tablespoon cooking oil
1 cups water 3 tablespoons yeast
1 tablespoon salt
Solution.

Question. 104 Many schools have a recommended students-teachers ratio as 35:1. Next year, school expects an increase in enrollment by 280 students. How many new teachers will they to appoint to maintain the students-teachers ratio?
Solution.

Question. 105 Kusum always forgets how to convert miles to kilometres and back again. However, she remembers that her car’s speedometer shows both miles and kilometres. She knows that travelling 50 miles per hour is same as travelling 80 kilometres per hour. To cover a distance of 200 km, how many miles Kusum would have to go?
Solution.

Question. 106 The student of Anju’s class sold posters to raise money. Anju wanted to create a ratio for finding the amount of money, her class would make for different numbers of posters sold. She knew, they could raise Rs 250 for every 60 posters sold.
(a) How much money would Anju’s class make for selling 102 posters?
(b) Could Anju’s class raise exactly Rs 2000? If so, how many posters would they need to sell? If not, why?
Solution.