# NCERT solutions for Mathematics Exemplar Class 8 chapter 10 - Direct and Inverse Proportions [Latest edition]

## Chapter 10: Direct and Inverse Proportions

Exercise
Exercise [Pages 315 - 330]

### NCERT solutions for Mathematics Exemplar Class 8 Chapter 10 Direct and Inverse Proportions Exercise [Pages 315 - 330]

#### Choose the correct alternative:

Exercise | Q 1 | Page 315

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?

• 2 and 3

• 8 and 12

• 15 and 2

• 25 and 37.5

Exercise | Q 2 | Page 315

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?

• 12 and 5

• 15 and 4

• 25 and 2.4

• 45 and 1.3

Exercise | Q 3 | Page 315

Assuming land to be uniformly fertile, the area of land and the yield on it vary ______.

• Directly with each other

• Inversely with each other

• Neither directly nor inversely with each other

• Sometimes directly and sometimes inversely with each other

Exercise | Q 4 | Page 315

The number of teeth and the age of a person vary ______.

• Directly with each other

• Inversely with each other

• Neither directly nor inversely with each other

• Sometimes directly and sometimes inversely with each other

Exercise | Q 5 | Page 316

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 ______.

• 100 litres

• 50 litres

• 25.16 litres

• 25 litres

Exercise | Q 6 | Page 316

By travelling at a speed of 48 kilometres per hour, 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 ______.

• 60 km/h

• 80 km/h

• 30 km/h

• 40 km/h

Exercise | Q 7 | Page 316

In which of the following case, do the quantities vary directly with each other?

•  x 0.5 2 8 32 y 2 8 32 128
•  p 12 22 32 42 q 13 23 33 43
•  r 2 5 10 25 50 s 25 10 5 2 0.5
•  u 2 4 6 9 12 v 18 9 6 4 3
Exercise | Q 8 | Page 316

Which quantities in the previous question vary inversely with each other?

• x and y

• p and q

• r and s

• u and v

Exercise | Q 9 | Page 316

Which of the following vary inversely with each other?

• Speed and distance covered

• Distance covered and taxi fare

• Distance travelled and time taken

• Speed and time taken

Exercise | Q 10 | Page 316

Both x and y are in direct proportion, then 1/x and 1/y are ______.

• In indirect proportion

• In inverse proportion

• Neither in direct nor in inverse proportion

• Sometimes in direct and sometimes in inverse proportion

Exercise | Q 11 | Page 317

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 ______.

• 20/3 km/h

• 16 km/h

• 20 km/h

• 15 km/h

Exercise | Q 12 | Page 317

100 persons had food provision for 24 days. If 20 persons left the place, the provision will last for ______.

• 30 days

• 96/5 days

• 120 days

• 40 days

Exercise | Q 13 | Page 317

If two quantities x and y vary directly with each other, then ______.

• x/y remains constant

• x – y remains constant

• x + y remains constant

• x × y remains constan

Exercise | Q 14 | Page 317

If two quantities p and q vary inversely with each other, then ______.

• p/q remains constant

• p + q remains constant

• p × q remains constant

• p – q remains constant

Exercise | Q 15 | Page 317

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 ______.

• 250/9 m

• 500/9 m

• 1000 m

• 500/3 m

Exercise | Q 16 | Page 318

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?

• 25 and 35

• 35 and 25

• 35 and 49

• 15 and 21

Exercise | Q 17 | Page 318

If x = 5y, then x and y vary ______ with each other.

Exercise | Q 18 | Page 318

If xy = 10, then x and y vary ______ with each other.

Exercise | Q 19 | Page 318

When two quantities x and y are in ______ proportion or vary ______ they are written as x oo y.

Exercise | Q 20 | Page 318

When two quantities x and y are in ______ proportion or vary ______ they are written as x oo 1/y.

Exercise | Q 21 | Page 318

Both x and y are said to vary ______ with each other if for some positive number k, xy = k.

Exercise | Q 22 | Page 318

x and y are said to vary directly with each other if for some positive number k, ______ = k.

Exercise | Q 23 | Page 318

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.

Exercise | Q 24 | Page 318

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.

Exercise | Q 25 | Page 318

If 12 pumps can empty a reservoir in 20 hours, then time required by 45 such pumps to empty the same reservoir is ______ hours.

Exercise | Q 26 | Page 319

If x varies inversely as y, then

 x – 60 y 2 10
Exercise | Q 27 | Page 319

If x varies directly as y, then

 x 12 6 y 48 –
Exercise | Q 28 | Page 319

When the speed remains constant, the distance travelled is ______ proportional to the time.

Exercise | Q 29 | Page 319

On increasing a, b increases in such a manner that a/b remains ______ and positive, then a and b are said to vary directly with each other.

Exercise | Q 30 | Page 319

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.

Exercise | Q 31 | Page 319

If two quantities x and y vary directly with each other, then ______ of their corresponding values remains constant.

Exercise | Q 32 | Page 319

If two quantities p and q vary inversely with each other then ______ of their corresponding values remains constant.

Exercise | Q 33 | Page 319

The perimeter of a circle and its diameter vary ______ with each other.

Exercise | Q 34 | Page 319

A car is travelling 48 km in one hour. The distance travelled by the car in 12 minutes is ______.

Exercise | Q 35 | Page 319

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 ______.

Exercise | Q 36 | Page 319

If the thickness of a pile of 12 cardboard sheets is 45 mm, then the thickness of a pile of 240 sheets is ______ cm.

Exercise | Q 37 | Page 319

If x varies inversely as y and x = 4 when y = 6, then when x = 3 the value of y is ______.

Exercise | Q 38 | Page 319

In direct proportion, a_1/b_1 ______ a_2/b_2

Exercise | Q 39 | Page 320

In case of inverse proportion, a_2/- = b_2/-

Exercise | Q 40 | Page 320

If the area occupied by 15 postal stamps is 60 cm2, then the area occupied by 120 such postal stamps will be ______.

Exercise | Q 41 | Page 320

If 45 persons can complete a work in 20 days, then the time taken by 75 persons will be ______ hours.

Exercise | Q 42 | Page 320

Devangi travels 50 m distance in 75 steps, then the distance travelled in 375 steps is ______ km.

#### State whether the following statement is True or False:

Exercise | Q 43 | Page 320

Two quantities x and y are said to vary directly with each other if for some rational number k, xy = k.

• True

• False

Exercise | Q 44 | Page 320

When the speed is kept fixed, time and distance vary inversely with each other.

• True

• False

Exercise | Q 45 | Page 320

When the distance is kept fixed, speed and time vary directly with each other.

• True

• False

Exercise | Q 46 | Page 320

Length of a side of a square and its area vary directly with each other

• True

• False

Exercise | Q 47 | Page 320

Length of a side of an equilateral triangle and its perimeter vary inversely with each other.

• True

• False

Exercise | Q 48 | Page 321

If d varies directly as t2, then we can write dt2 = k, where k is some constant.

• True

• False

Exercise | Q 49 | Page 321

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.

• True

• False

Exercise | Q 50 | Page 321

If x and y are in direct proportion, then (x – 1) and (y – 1) are also in direct proportion.

• True

• False

Exercise | Q 51 | Page 321

If x and y are in inverse proportion, then (x + 1) and (y + 1) are also in inverse proportion.

• True

• False

Exercise | Q 52 | Page 321

If p and q are in inverse variation then (p + 2) and (q – 2) are also in inverse proportion.

• True

• False

Exercise | Q 53 | Page 321

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

• True

• False

Exercise | Q 54 | Page 321

When two quantities are related in such a manner that, if one increases, the other also increases, then they always vary directly

• True

• False

Exercise | Q 55 | Page 321

When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely.

• True

• False

Exercise | Q 56 | Page 321

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

• True

• False

Exercise | Q 57 | Page 321

The number of workers and the time to complete a job is a case of direct proportion.

• True

• False

Exercise | Q 58 | Page 321

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

• True

• False

Exercise | Q 59 | Page 321

The area of cultivated land and the crop harvested is a case of direct proportion.

• True

• False

Exercise | Q 60.(i) | Page 321

Which of the following vary directly and which vary inversely with other and which are neither of the two?

The time taken by a train to cover a fixed distance and the speed of the train.

Exercise | Q 60.(ii) | Page 321

Which of the following vary directly and which vary inversely with other and which are neither of the two?

The distance travelled by CNG bus and the amount of CNG used.

Exercise | Q 60.(iii) | Page 321

Which of the following vary directly and which vary inversely with other and which are neither of the two?

The number of people working and the time to complete a given work.

Exercise | Q 60.(iv) | Page 321

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Income tax and the income.

Exercise | Q 60.(v) | Page 321

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Distance travelled by an auto-rickshaw and time taken.

Exercise | Q 61.(i) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Number of students in a hostel and consumption of food.

Exercise | Q 61.(ii) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Area of the walls of a room and the cost of white washing the walls.

Exercise | Q 61.(iii) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

The number of people working and the quantity of work.

Exercise | Q 61.(iv) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Simple interest on a given sum and the rate of interest.

Exercise | Q 61.(v) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Compound interest on a given sum and the sum invested.

Exercise | Q 62.(i) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

The quantity of rice and its cost.

Exercise | Q 62.(ii) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

The height of a tree and the number of years.

Exercise | Q 62.(iii) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Increase in cost and number of shirts that can be purchased if the budget remains the same.

Exercise | Q 62.(iv) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Area of land and its cost.

Exercise | Q 62.(v) | Page 322

Which of the following vary directly and which vary inversely with other and which are neither of the two?

Sales Tax and the amount of the bill.

#### Solve the following:

Exercise | Q 63 | Page 322

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

Exercise | Q 64 | Page 322

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

Exercise | Q 65 | Page 322

l varies directly as m and l is equal to 5, when m = 2/3. Find l when m = 16/3

Exercise | Q 66 | Page 322

If x varies inversely as y and y = 60 when x = 1.5. Find x. when y = 4.5.

Exercise | Q 67 | Page 322

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?

Exercise | Q 68 | Page 322

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?

Exercise | Q 69 | Page 322

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

Exercise | Q 70 | Page 323

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?

Exercise | Q 71.(i) | Page 323

It is given that l varies directly as m. Write an equation which relates l and m.

Exercise | Q 71.(ii) | Page 323

It is given that l varies directly as m. Find the constant of proportion (k), when l is 6 then m is 18.

Exercise | Q 71.(iii) | Page 323

It is given that l varies directly as m. Find l, when m is 33.

Exercise | Q 71.(iv) | Page 323

It is given that l varies directly as m. Find m when l is 8.

Exercise | Q 72 | Page 323

If a deposit of Rs 2,000 earns an interest of Rs 500 in 3 years, how much interest would a deposit of Rs 36,000 earn in 3 years with the same rate of simple interest?

Exercise | Q 73 | Page 323

The mass of an aluminum 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.

Exercise | Q 74 | Page 323

Find the values of x and y if a and b are in inverse proportion:

a. 12 x 8
b. 30 5 y

Exercise | Q 75 | Page 323

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

Exercise | Q 76 | Page 323

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?

Exercise | Q 77.(i) | Page 323

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

 x 3 6 15 20 30 y 12 24 45 60 120
Exercise | Q 77.(ii) | Page 323

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

 x 4 7 10 10 y 24 42 60 96
Exercise | Q 77.(iii) | Page 323

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

 x 1 4 9 20 y 1.5 6 13.5 30
Exercise | Q 78.(i) | Page 324

If a and b vary inversely to other, then find the values of p, q, r;

x, y, z and l, m, n.

 a 6 8 q 25 b 18 p 39 r
Exercise | Q 78.(ii) | Page 324

If a and b vary inversely to other, then find the values of p, q, r;

x, y, z and l, m, n.

 a 2 y 6 10 b x 12.5 15 z
Exercise | Q 78.(iii) | Page 324

If a and b vary inversely to other, then find the values of p, q, r;

x, y, z and l, m, n.

 a l 9 n 6 b 5 m 25 10
Exercise | Q 79.(i) | Page 324

If 25 metres of cloth costs Rs 337.50, then what will be the cost of 40 metres of the same type of cloth?

Exercise | Q 79.(ii) | Page 324

If 25 metres of cloth costs Rs 337.50, then what will be the length of the cloth bought for Rs 810?

Exercise | Q 80 | Page 324

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 2/3 hours?

Exercise | Q 81 | Page 324

The cost of 27 kg of iron is Rs 1,080, what will be the cost of 120 kg of iron of the same quality?

Exercise | Q 82 | Page 324

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?

Exercise | Q 83 | Page 324

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?

Exercise | Q 84.(i) | Page 324

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 darkest blue shade?

Exercise | Q 84.(ii) | Page 324

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 darkest blue shade?

Exercise | Q 84.(iii) | Page 324

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 darkest blue shade?

Exercise | Q 84.(iv) | Page 324

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 darkest blue shade

Exercise | Q 85 | Page 325

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

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

Exercise | Q 86 | Page 325

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?

Exercise | Q 87 | Page 326

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

Exercise | Q 88 | Page 326

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

Exercise | Q 89 | Page 326

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?

Exercise | Q 90 | Page 326

Ravi starts for his school at 8:20 a.m. on his bicycle. If he travels at a speed of 10km/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?

#### Match each of the entries in Column I with the appropriate entry in Column II

Exercise | Q 91 | Page 326
 Column I Column II 1. x and y vary inversely to each other A. x/y = Constant 2. Mathematical representation of inverse variation of quantities p and q B. y will increase in proportion 3. Mathematical representation of direct variation of quantities m and n C. xy = Constant 4. When x = 5, y = 2.5 and when y = 5, x = 10 D. p  oo  1/q 5. When x = 10 , y = 5 and when x = 20, y = 2.5 E. y will decrease in proportion 6. x and y vary directly with each other F. x and y are directly proportional 7. If x and y vary inversely then on decreasing x G. m  oo  n 8. If x and y vary directly then on decreasing x H. x and y vary inversely I. p  oo  q J. m  oo  1/n
Exercise | Q 92 | Page 326

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

Exercise | Q 93 | Page 326

Ms. Anita has to drive from Jhareda to Ganwari. She measures a distance of 3.5 cm between these villages on the map. What is the actual distance between the villages if the map scale is 1 cm = 10 km?

Exercise | Q 94 | Page 327

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 lengths of the shadows are directly proportional to their heights. Find the height of the tank.

Exercise | Q 95 | Page 327

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

 Distance (m) Time (sec.) Elevator- A 435 39 Elevator- B 448 28 Elevator- C 130 10 Elevator- D 85 5

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

Exercise | Q 96 | Page 327

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.

Exercise | Q 97 | Page 328

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?

Exercise | Q 98 | Page 328

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

Exercise | Q 99 | Page 328

A bowler throws a cricket ball at a speed of 120 km/h. How long does this ball take to travel a distance of 20 metres to reach the batsman?

Exercise | Q 100 | Page 328

The variable x is inversely proportional to y. If x increases by p%, then by what percent will y decrease?

Exercise | Q 101.(a) | Page 328

Here is a keyboard of a harmonium: Find the ratio of white keys to black keys on the keyboard

Exercise | Q 101.(b) | Page 328

Here is a keyboard of a harmonium: What is the ratio of black keys to all keys on the given keyboard.

Exercise | Q 101.(c) | Page 328

Here is a keyboard of a harmonium: 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.

Exercise | Q 102 | Page 329

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

 Litres of gas 1 0.5 2 2.5 3 5 Distance (km) 15 7.5 30 37.5 45 75

Which type of properties are indicated by the table? How much distance will be covered by the car in 8 litres of gas?

Exercise | Q 103 | Page 329

Kritika is following this recipe for bread. She realises her sister used most of sugar syrup for her breakfast. Kritika has only 1 6 cup of syrup, so she decides to make a small size of bread. How much of each ingredient shall she use?
1 cup quick cooking oats
1/3 cup sugar syrup
1 tablespoon cooking oil
1 1/3 cups water
3 tablespoons yeast
1 teaspoon salt.

Exercise | Q 104 | Page 329

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

Exercise | Q 105 | Page 329

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?

Exercise | Q 106.(a) | Page 330

The students 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. How much money would Anju’s class make for selling 102 posters?

Exercise | Q 106.(b) | Page 330

The students 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. Could Anju’s class raise exactly Rs 2,000? If so, how many posters would they need to sell? If not, why?

Exercise

## NCERT solutions for Mathematics Exemplar Class 8 chapter 10 - Direct and Inverse Proportions

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Concepts covered in Mathematics Exemplar Class 8 chapter 10 Direct and Inverse Proportions are Concept of Direct Proportion, Concept of Inverse Proportion.

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