Ratio And Proportion

Ratio is given by

Hence ratio is 2:5

Number of sides in a square = 4

Number of edges in a cube = 12

Ratio of sides to edges is given by

Hence the ratio is 1:3

Width = 60cm

Length = 180cm

Perimeter = 2(l+b) = 2 (180+60) = 480 cm

Ratio of width to perimeter =

Hence the ratio is 1:8

Savings= Rs 36000


Expenditure = 288000-36000

= 252000








Hence ratio is 1:7.

From page 261 to page 272 including first and last page

Number of pages = 12

Hence ratio =

The ratio of the number of pages of this chapter to the total number of pages of the book is 3:80

As the simplest form of the ratio is 7:4

Therefore the common factor of 7 and 4 can be taken as x.

Therefore total number of marbles is 7x+4x = 11x.

Therefore number of marbles in the box is a multiple of 11.
As 11x2 = 22

Therefore number of marbles = 22

As the simplest form of the ratio is 2:3

Therefore the common factor of 2 and 3 can be taken as x.

Therefore 2x = 18

x = 9

Therefore number of books with brown cover = 3x9 = 27

Between

The greater fraction is

Between

Between

Hence the greatest is 40:25

Total number of students in the class are (b + g)

Hence ratio is

As speed =

Speed of bus = 40km/hr

Speed of train = 64km/hr

Distance travelled in 1 hr by bus = 40km

Distance travelled in 1 hr by train = 64km

Ratio of distance travelled by bus to train =

In questions 11 to 15, find the missing number in the box in each of the proportions:

Let blank = x

in the first blank

In the second blank

In the first blank

In the second blank

In the third blank

TRUE

Simplified state of

Hence TRUE

TRUE

Simplified state of

Hence TRUE

FALSE

Hence FALSE

FALSE

Hence FALSE

FALSE

Hence FALSE

TRUE

Hence TRUE

FALSE

As units are equal they get cancelled

And

Hence FALSE

TRUE

As units are equal they get cancelled

And

1 metre = 100cm hence

Hence TRUE

TRUE

As units are equal they get cancelled

And

Hence TRUE

TRUE

As units are equal they get cancelled

Hence ratio = 1:10

Hence TRUE.

As 1m = 100cm

Ratio of 150 cm to 1m is

=

Making the ratio 1:1.5 in whole numbers we multiply and divide it by 2 which gives us

1:1.5 = 2:3

Hence TRUE

1kg = 1000g

25kg = 25000g

25kg: 20g = 25000g:20g = 2500:2 = 1250:1

50kg = 50000g

50kg:40g = 50000g:40g = 5000:4 = 2000:2 = 1000:1

Hence FALSE

Number of hours in 1 day = 24 hours

1 hour: 1 day = 1hour: 24 hours = 1:24

Hence FALSE

Hence 4:16 in its lowest form is 1:4

Hence FALSE

ratio 5:4 =

Ratio 4:5 =

As

Hence they are different.

Hence TRUE

A ratio can be any rational number more than or less than or equal to 1.

Hence FALSE

TRUE

A ratio can be equal to 1.

Taking reciprocal

Hence they are not in proportion.

Hence FALSE

FALSE.

The ratio cannot be in two different units.

Division.

x =

If the number in the box is x and x,24,9,12 are in proportion then,

proportion.

Use Fig. 8.2 (In which each square is of unit length) for questions 39 and 40:

Perimeter of shaded area = 2 + 1 + 1 + 2 = 6 units

Perimeter of whole figure = 3 + 3 + 4 + 4 = 14units

Ratio =

Area of shaded part = 2 × 1 = 2cm²

Area of whole figure = 4 × 3 = 12cm²

Ratio =

Sleeping time = 18 hours

Awaking time = 24 - 18 = 6 hours

Ratio =

1

Same

5paise:25paise = 20 paise : x paise

5:25 = 20: x

As 1 hour = 60mins

Time taken by saturn = (9x60) + 56 mins = 596 mins

Time taken by jupiter = (10x60) + 40 = 640 mins

Ratio =

10g: 100mL = x: 1litre

As 1 litre = 1000mL

10g:100mL = x:1000mL

Marked price = Rs 625

Sale price = Rs 500


ratio =

(i) Simplest form of

Hence the two ratios are equal

(ii) Simplest form of

Hence the two ratios are equal

(iii)

_{Simplest form of}

Hence the two ratios are not equal

and

= 10 × 93 and 21 × 21

= 930 and 441

Hence 10:21 is larger

let the common factor of 7 and 2 be x.

Therefore 7x + 2x = 18kg

9x = 18kg

X = 2kg

Therefore amount of khoya used = 7x2 = 14kg.

Consider AB as the segment having length 56 cm

Let C be the point which divides the segment AB in ratio 2:5 and the two parts as AC and CB

Let AC = a, from figure we have AB = AC + CB

⇒ 56 = a + CB

⇒ CB = 56 – a

Now given is AC:BC = 2:5

⇒ =

⇒ =

cross multiply

⇒ (56 – a) × 2 = 5 × a

⇒ 112 – 2a = 5a

⇒ 112 = 5a + 2a

⇒ 112 = 7a

⇒ a =

⇒ a = 16

Therefore AC = a = 16 cm and CB = 56 – a = 56 – 16 = 40 cm

Number of milk teeth = 20

Number of permanent teeth = 32

Ratio of number of milk teeth to the number of permanent teeth is given as

Ratio =

⇒ Ratio =

Divide numerator and denominator by 4 we get

⇒ Ratio =

Therefore, ratio of number of milk teeth to the number of permanent teeth is 5:8

There are 3732 females per 4000 males

Therefore, to find number of females per 1000 males we need to divide 3732 by 4

⇒ number of females per 1000 males = = 933

Therefore, sex ratio = = 0.933

Income = 360000 Rs

Income tax = 24000 Rs

(a) ratio of income to income tax =

⇒ ratio = = = 15

⇒ ratio = 15:1

(b) income after paying income tax = income – income tax

= 360000 – 24000

= 336000 Rs

Ratio of income tax to income after paying income tax =

⇒ ratio = =

Dividing numerator and denominator by 24 we get

⇒ ratio =

Therefore, ratio of income tax to income after paying income tax is 1:14

Ramesh’s earnings = 28000 Rs

Rama’s earnings = 36000 Rs

Total earnings = Ramesh’s earnings + Rama’s earnings

= 28000 + 36000

= 64000 Rs

(a) ratio of Ramesh’s earnings to their total earnings =

⇒ ratio = =

Divide the numerator and the denominator by 4 we get

⇒ ratio =

Therefore, ratio of Ramesh’s earnings to their total earnings is 7:16

(b) ratio of Rama’s earnings to their total earnings =

⇒ ratio = =

Divide the numerator and the denominator by 4 we get

⇒ ratio =

Therefore, ratio of Rama’s earnings to their total earnings is 9:16

Number of men = 112

Total number of persons = 288

Therefore, number of women = Total number of persons - Number of men

number of women = 288 – 112 = 176

(a) ratio of number of men to that of women =

⇒ ratio =

Dividing numerator and denominator by 2 we get

⇒ ratio =

Again dividing the numerator and denominator by 8 we get

⇒ ratio =

Therefore, ratio of number of men to that of women = 7:11

(b) ratio of number of men to total number of persons =

⇒ ratio =

Dividing numerator and denominator by 4 we get

⇒ ratio =

Again dividing the numerator and denominator by 8 we get

⇒ ratio =

Therefore, ratio of number of men to total number of persons = 7:18

(c) ratio of women to the total number of persons =

⇒ ratio =

Dividing numerator and denominator by 4 we get

⇒ ratio =

Again dividing the numerator and denominator by 8 we get

⇒ ratio =

Therefore, ratio of number of women to total number of persons = 11:18

Length of paper = 1.2 m

Width of paper = 21 cm

As the units are different we cannot directly take the ratio to take ratio we need to have two quantities with same units

1 m = 100 cm

⇒ 1.2 m = 1.2 × 100 = 120 cm

∴ Length of paper = 120 cm

ratio of width of the paper to its length =

⇒ ratio =

Divide numerator and denominator by 3 we get

⇒ ratio =

Therefore, ratio of width of the paper to its length is 7:40

Speed =

Distance covered by scooter = 120 km

Time taken by scooter = 3 hours

Speed of scooter =

⇒ speed of scooter = = 40 km/hour

Distance covered by train = 120 km

Time taken by train = 2 hours

Speed of train =

⇒ speed of train = = 60 km/hour

Ratio of their speed =

⇒ Ratio of their speed =

Divide numerator and denominator by 20

⇒ Ratio of their speed =

Lunch break period = 30 minutes

9 a.m. to 5.30 p.m. is 7 hours and 30 minutes

1 hour is 60 minutes

⇒ 7 hours = 7 × 60 = 540 minutes

Total period in office = 540 + 30 = 570 minutes

Therefore, ratio of lunch break to the total period in the office =

⇒ ratio = =

Dividing numerator and denominator by 3 we get

⇒ ratio =

Therefore, ratio of lunch break to the total period in the office is 1:18

The ratio of shadow to the height would remain the same

When height is 3 m the shadow is 4 m

⇒ ratio = = …(i)

When shadow is 24 m let the height be ‘a’ m

⇒ ratio = =

Using (i)

⇒ =

Cross multiplying

⇒ 24 × 3 = 4 × a

⇒ a = = 6 × 3

⇒ a = 18

Height of the flagstaff is 18 m

The ratio of number of cups of milk to number of people is constant

When number of cups of milk = 1 number of people are 6

⇒ ratio = = …(i)

When number of people are 8 let number of cups of milk required be ‘a’

⇒ ratio = =

Using (i)

⇒ =

Cross multiply

⇒ a =

Divide numerator and denominator by 2 we get

⇒ a = = = +

⇒ a = 1 + = 1 cups of milk

The ratio number of cups of milk to number of cups of flour is constant

When number of cups of milk is 1 number of cups of flour =

⇒ ratio = = = …(ii)

Let number of cups of flour required be ‘b’ when number of cups of milk are

⇒ ratio = = =

Using (ii)

⇒ =

Cross multiply

⇒ 3b × 2 = 4 × 5

⇒ b =

Divide numerator and denominator by 2

⇒ b = = = +

⇒ b = 3 cups of flour

Therefore, cups of milk needed to make cake for 8 people is 1 and that of flour is 3

Ratio of number of large classrooms to small classrooms = 3:4

Number of small classrooms = 20

⇒ ratio = =

⇒ =

Cross multiply we get

⇒ Number of large classrooms = = 3 × 10

Therefore, number of large classrooms is 30

Total newspapers = 312

English newspapers = 216

Hindi newspapers = Total newspapers - English newspapers

⇒ Hindi newspapers = 312 – 216 = 96

(a) ratio of number of English newspapers to the number of Hindi newspapers =

⇒ ratio =

Divide numerator and denominator by 3 gives

⇒ ratio =

Divide numerator and denominator by 8 gives

⇒ ratio =

Therefore, ratio of number of English newspapers to the number of Hindi newspapers is 9:4

(b) ratio of number of Hindi newspapers to the total number of newspapers =

⇒ ratio =

Divide numerator and denominator by 3 gives

⇒ ratio =

Divide numerator and denominator by 8 gives

⇒ ratio =

Therefore, ratio of number of Hindi newspapers to the total number of newspapers is 4:13

Hindu students = 288

Muslim students = 252

Christian students = 72

Total students = Hindu students + Muslim students + Christian students

Total students = 288 + 252 + 72 = 612

(a) ratio of number of Hindu students to the number of Christian students =

⇒ ratio =

Divide numerator and denominator by 72

⇒ ratio =

Therefore, ratio of number of Hindu students to the number of Christian students is 4:1

(b) ratio of number of Muslim students to the total number of students =

⇒ ratio =

Divide numerator and denominator by 6 we get

⇒ ratio =

Divide numerator and denominator by 6 we get

⇒ ratio =

Therefore, ratio of number of Muslim students to the total number of students is 7:17

Total stalls = 117

Let the number of north Indian stalls be ‘a’

So, the number of south Indian stalls = 117 – a

ratio of North Indian food stalls to South Indian food stalls = =

⇒ =

Cross multiply we get

⇒ 4a = 5 × (117 – a)

⇒ 4a = 585 – 5a

⇒ 4a + 5a = 585

⇒ 9a = 585

⇒ a =

∴ a = 65

∴ 117 – a = 117 – 65 = 52

Number of north Indian stalls is 65 and number of south Indian stalls is 52

Number of cycles = 115

Number of scooters = 75

Number of bikes = 45

Total number of vehicles = Number of cycles + Number of scooters + Number of bikes

Total number of vehicles = 115 +75 + 45 = 235

ratio of the number of cycles to the total number of vehicles =

⇒ ratio =

Divide numerator and denominator by 5 we get

⇒ ratio =

Therefore, ratio of the number of cycles to the total number of vehicles is 23:47

Speed = …(i)

Distance covered Ajmer to Jaipur = 130 km

Time taken = 2 hours

Using (i)

Speed = = 65 km/hour …(ii)

Speed is uniform

Let the time taken to cover distance between Delhi and Bhopal be ‘a’

Distance between Delhi and Bhopal = 780 km

Using (i) and (ii)

⇒ 65 =

⇒ a =

Divide numerator and denominator by 5 we get

⇒ a = = 12 hours

Time to travel from Delhi to Bhopal is 12 hours

If the ratio of length to breadth of both school ground and mela are equal then they are said to be in proportion

Length of school ground = 150 m

Breadth of school ground = 90 m

⇒ ratio = =

Divide the numerator and denominator by 30

⇒ ratio of length to breadth of school ground = …(i)

Length of mela = 210 m

Breadth of mela= 126 m

⇒ ratio = =

Divide the numerator and denominator by 6

⇒ ratio =

Divide the numerator and denominator by 7

⇒ ratio of length to breadth of mela = …(ii)

From (i) and (ii) we can say that measurements are in proportion

By counting the number of squares under each continent we can get to know the areas as follows

Africa = 26 sq. units

Europe = 10 sq. units

Australia = 8 sq. units

Asia = 44 sq. units

Antarctica = 13 sq. units

North America = 17 sq. units

South America = 18 sq. units

(a) ratio of area of Africa to Europe = =

Divide numerator and denominator by 2 we get

ratio of area of Africa to Europe = = 13:5

(b) ratio of area of Australia to Asia = =

Divide numerator and denominator by 4 we get

ratio of area of Australia to Asia = = 2:11

(c) ratio of area of Antarctica to combined area of North America and South America = = = = 13:35

Let the two varieties of tea be ‘A’ and ‘B’

Cost of ‘A’ = 234 Rs

Cost of ‘B’ = 130 Rs

Ratio of their costs = =

Divide numerator and denominator by 2

⇒ Ratio of their costs =

Divide numerator and denominator by 13

⇒ Ratio of their costs = …(i)

Weight of mixture = 84 kg

Let the weight of type ‘A’ tea in the mixture be ‘a’ and as total is 84 the weight of type ‘B’ tea would be (84 – a)

Now given that the ratio of two varieties of tea in the mixture is same as the ratio of their costs

Using (i) and given condition

⇒ =

Cross multiply

⇒ 9 × (84 – a) = 5a

⇒ 756 – 9a = 5a

⇒ 756 = 14a

⇒ a =

Divide numerator and denominator by 7 we get

⇒ a = = 54 kg

⇒ 84 – a = 84 – 54 = 30 kg

Therefore, weight of 234 Rs/kg tea in mixture is 54 kg and weight of 130 Rs/kg tea in mixture is 30 kg

Let the weight of zinc and copper in the alloy be the 7x and 9x

Weight of the alloy = 8 kg

Total weight = 7x+9x = 16x

16x = 8

Weight of copper = 9× = 4.5 kg

Hence, weight of the copper in alloy 4.5 kg.

From given figure

AC= 2

AF = 5

AG = 2

AD = 1

BF =1

AI = 2

CE =2

DI = 5

(i) AC:AF = 2:5

(ii) AG:AD = 2:1

(iii) BF:AI = 1:2

(iv) CE:DI = 2:5

Let two numbers be 9x and 16x

9x+16x = 100

25x=100

First number = 9× 4 = 36

Second number = 16× 4 = 64

Hence, two number be 36 and 64.

From fig (i)

Area of shaded portion = 8 sq. unit

Area of whole figure = 16 sq. unit

Ratio of area of shaded portion to whole figure =

Ratio = 1:2

From figure (ii)

Area of shaded portion = 8 sq. unit

Area of whole figure = 16 sq. unit

Ratio of area of shaded portion to whole figure =

Ratio = 1:2

Total pages of manuscript to type = 40

Total typed pages = 30

Total left pages = 40 – 30 = 10

Ratio of the number of pages typed to the number of pages left =

Ratio = 3:1

Length of the tiles is 40 cm.

∴ One part of length =

Width of the tiles = 60 cm

One part of width =

There are two part of length and three parts of width.

Length of shaded portion = 2× 10 = 20 cm

Width of shaded portion = 3× 12 = 36 cm

a) Perimeter of shaded portion = 2(20+36) = 2× 56 = 112 cm

Perimeter of whole design = 2(40+60) = 2× 100 = 200 cm

Ratio of the perimeter of shaded portion to the perimeter of the whole design =

Ratio = 14:25

b) Area of shaded portion = 20× 36 = 720 sq. cm

Area of unshaded portion = area of whole figure – area of shaded portion

= 40× 60 – 720

= 2400 – 720 = 1680 sq. cm

Ratio of the area of the shaded portion to the area of the unshaded portion =

Ratio = 3:7

From the given figure,

Area of shaded portion I = length× width

= 5× 5

= 25 sq. units

Area of portion III = 5× 7 = 35 sq. units

Area of shaded portion II = area of whole portion – (area of shaded portion I+ area of portion III)

= 10× 10 – (25+35)

= 100 – 60 = 40 sq. unit

Area of shaded portions I and II taken together = 25+40 = 65 sq. unit

a) Ratio of area of shaded portion I to shaded portion II=

Ratio = 5:8

b) Ratio of area of shaded portion II to shaded portion III =

Ratio = 8:7

c) Ratio of the area of shaded portions I and II taken together and shaded portion III =

Ratio = 13:7

Distance travelled by car in 15 litres = 240 km

Distance travelled by car in 1 litres = =16 km

∴ distance travelled by car in 25 litres = 16× 25 = 400 km

Hence, Car will travel 400 km in 25 liters of patrol.

Earning of Bachhu Manjhi in 8 months = Rs. 24,000

Earning of Bachhu Manjhi in 1 months = = Rs. 3,000

a) Earning of Bachhu Manjhi in one year = 3000× 12 = Rs. 36,000 (∵ 1 year = 12 months)

b) Let after ‘x’ months he will earn 42,000

Earning of Bachhu Manjhi in x months = 3000x

3000x=42000

Hence, after 14 months Bachhu Manjhi will earn 42,000

360 quintals wheat is yielded by 8 hectares.

1 quintals wheat is yielded by = hectares

∴ 540 quintals wheat is yielded by = =12 hectares.

Hence, 540 quintals will be yielded by 12 hectares.

Earth rotates in 24 hours = 360°

Earth will rotate in 1 hour = = 15°

Earth will rotate in 2 hour = 15× 2 = 30°

Hence, earth will rotate 30° in 2 hour.

Number of iron tablets Shivani has to taken in one day = 2

Total number of tablets in 15 days = 15× 2 = 30

Cost of 10 tablets = 17

Cost of 1 tablets =

∴ cost of 30 tablets =

Hence, cost for her medical bill for 15 days = Rs. 51

One quarterly = 3 months.

Quarterly fee = Rs. 540

3 months fee = Rs. 540

1 month fee = = Rs. 180

Fee for 7 months = 7× 180 = Rs. 1260

Hence, fee for seven months will Rs. 1260

Let the vote cast for two candidates be 5x and 7x

Successful candidate will receive greater votes.

∴ 7x = 20734

x = 2962

opponent votes = 5x = 5× 2962 = 14810

Hence, opponent will receive 14810 votes.

Weight of 3 meter long pipe = 7.6 kg

Weight of 1 meter pipe = kg

Weight of 7.8 meter long pipe = kg

Hence, weight of 7.8m long pipe will 19.76 kg

For recipe of raspberry jelly,

5 cups of raspberry juice = sugar needed cups = cups

For 1 cup of raspberry juice, sugar needed = cup

∴ for 6 cups of the juice, sugar needed = cups

Hence, 3 cups of sugar needed for 6 cups of raspberry juice.

Total plant, planted by farmer = 1890

Plants in each row = 63

∴ Number of rows =

Worm destroys 18 plants in each row,

∴ Total plants destroys by worm = 18× 30 = 540

Hence, worn will destroy 540 plants.

Length of the floor = 5 m

Width of the floor = 3 m

Area of the Room = length× width

Area of the room = 5× 3 = 15 m²

Area of one tile =

Area of 40 tiles =

Total 2.5 m² area will be covered by tiles.

Area not covered by tile = (15 – 2.5) = 12.5 m²

Ratio of the tiled and the non-tiled portion of the floor

Ratio = 1:5

Area of board = length× breadth

= 3× 2 = 6 m²

She cut rectangular piece = 250cm× 90cm

Area of rectangular piece = 250× 90 = 22500 cm²

∵

Area of rectangular piece =

Remaining area of board = 6 – 2.25 = 3.75 m²

Ratio of the area of cut out piece and the remaining piece =

=

Ratio = 3:5