NCERT Exemplar Solutions: Ratio & Proportion | Mathematics (Maths) Class 6 PDF Download

Ans: (a)
The comparison of two numbers or quantities by division is known as the ratio. Symbol ‘:’ is used to denote ratio.
Ratio of 8 books to 20 books = 8/20
Divide both numerator and denominator by 4.
= 2/5
Therefore, ratio of 8 books to 20 books = 2 : 5

Ans: (d)
We know that, number of sides in a square = 4 and number of edges in a cube = 12
So, ratio of sides to edges = 4/12
Divide both numerator and denominator by 4.
= 1/3
Therefore, ratio of sides to edges = 1 : 3

Ans: (d)
From the question it is given that,
Width of a picture = 60 cm
Length of a picture = 1.8 m
We know that, 1 m = 100 cm
So, 1.8 m = 180 cm
Perimeter of rectangle = 2 (length + breadth)
= 2 (180 + 60)
= 2 (240)
= 480
Therefore, The ratio of its width to its perimeter in lowest form = 60/480
Divide both numerator and denominator by 20.
= 3/24
Again, divide both numerator and denominator by 3.
= 1/8
= 1 : 8

Ans: (b)
From the question it is given that,
Neelam’s annual income is ₹ 288000
Her annual savings amount to ₹ 36000
So, Neelam’s expenditure = 288000 – 36000
= ₹ 252000
Then, ratio of her savings to her expenditure = 36000/252000
= 36/252
Divide both numerator and denominator by 12.
= 3/21
Again, divide both numerator and denominator by 3.
= 1/7
Therefore, ratio of her savings to her expenditure = 1 : 7

Ans: (c)
From the question it is given that,
Total number of pages in the Mathematics textbook for Class VI = 320 pages
The chapter ‘symmetry’ runs from page 261 to page 272
Number of pages contains symmetry chapter = 12
So, the ratio of the number of pages of symmetry chapter to the total number of pages of the book is,
= 12/320
Divide both numerator and denominator by 2.
= 6/160
Again, divide both numerator and denominator by 2.
= 3/80
Therefore, the ratio of the number of pages of this chapter to the total number of pages of the book is 3: 80.

Ans: (d)
From the question it is given that, the ratio of red marbles to blue marbles is 7:4.
Now, let us assume the common factor of 7 and 4 be x.
So, the total number of marbles in the box = 7x + 4x = 11x
Hence number of marbles in the box is a multiple of 11.
Therefore, 11 × 2 = 22

Ans: (c)
From the question it is given that,
On a shelf, books with green cover and that with brown cover are in the ratio 2:3
There are 18 books with green cover
So, let us assume the common factor of 2 and 3 be x.
Then, 2x = 18
x = 18/2
Divide both numerator and denominator by 2.
x = 9
Therefore, the number of books with brown cover is = 3x = 3 × 9
= 27

Ans: (d)
Consider the given ratios, 2 : 3, 5 : 8, 75 : 121 and 40 : 25.
Simplified form of 2: 3 = 2/3 = 0.67
Simplified form of 5 : 8 = 5/8 = 0.625
Simplified form of 75: 121 = 75/121 = 0.61
Simplified form of 40: 25 = 40/25 = 1.6
Therefore, the greatest ratio among the given ratios is 40 : 25

Ans: (a)
From the question,
Number of boys in the class = b
Number of girls in the class = g
Total number of students in the class = b + g
Therefore, the ratio of the number of boys to the total number of students in the class
= b/(b + g)

Ans: (c)
From the question it is given that,
Bus travels 160 km in 4 hours
Train travels 320 km in 5 hours
So, distance travelled by bus in an hour = 160/4 = 40 km/h
Distance travelled by train in an hour = 320/5 = 64 km/h
Then the ratio of the distances travelled by them in one hour is = 40/64
Divide both numerator and denominator by 8.
= 5/8
Therefore, the ratio of the distances travelled by them in one hour is 5: 8.

Let us assume the missing number be y.
Then, (3/5) = (y/20)
By cross multiplication we get,
(3 × 20)/5 = y
y = 60/5
Divide both numerator and denominator by 5.
y = 12
Therefore, 3/5 = [12]/20

Let us assume the missing number be y.
Then, y/18 = 2/9
By cross multiplication we get,
y = (2 × 18)/9
y = 36/9
Divide both numerator and denominator by 9.
y = 4
Therefore, [4]/18 = 2/9

Let us assume the missing number be y.
Then, 8/y = 3.2/4
By cross multiplication we get,
y = (8 × 4)/3.2
y = 32/3.2
y = 320/32
Divide both numerator and denominator by 32.
y = 10
Therefore, 8/[10] = 3.2/4

Consider the first two ratios [ ]/45 = 16/40
Let us assume the missing number be P
Then, P/45 = 16/40
By cross multiplication we get,
P = (16 × 45)/40
P = 720/40
P = 72/4
Divide both numerator and denominator by 4.
P = 18
Therefore, [18]/45 = 16/40
Now, consider the last two ratios, 16/40 = 24/[ ]
Let us assume the missing number be Q,
Then, 16/40 = 24/Q
By cross multiplication we get,
Q = (24 × 40)/16
Q = 960/16
Divide both numerator and denominator by 16.
Q = 60
Therefore, 16/40 = 24/[60]

Consider the first two ratios 16/36 = [ ]/63
Let us assume the missing number be P
Then, 16/36 = P/63
By cross multiplication we get,
P = (16 × 63)/36
P = 1008/36
Divide both numerator and denominator by 36.
P = 28
Therefore, 16/36 = [28]/63
Now, consider the middle two ratios, 28/63 = 36/[ ]
Let us assume the missing number be Q,
Then, 28/63 = 36/Q
By cross multiplication we get,
Q = (36 × 63)/28
Q = 2268/28
Divide both numerator and denominator by 28.
Q = 81
Therefore, 28/63 = 36/[81]
Consider the last two ratios 36/81 = [ ]/117
Let us assume the missing number be R
Then, 36/81 = R/117
By cross multiplication we get,
P = (36 × 117)/81
P = 4212/81
Divide both numerator and denominator by 81.
P = 52
Therefore, 36/81 = [52]/117
So, 16/36 = [28]/63 = 36/[81] = [52]/117

True.

Consider the two fractions, 3/8 = 15/40
15/40 is further simplified by dividing both numerator and denominator by 5 we get,
= 3/8
Therefore, 3/8 = 3/8

True.

False.

False.

False.

True

False.

True

True.

False

True.

False

5: 4 ≠ 4: 5
5/4 ≠ 4/5
1.25 ≠ 0.8

False.

False

False.

20m : 70m = Rs 8 : Rs 28.
Let us assume the missing number be P.
Then, 20m : 70m = ₹ 8 : ₹ P
20/70 = 8/P
P = (70 × 8)/20
P = 560/20
P = 56/2
P = 28
Therefore, 20m : 70m = Rs 8 : Rs 28.

There is a number in the box [ ] such that [ ], 24, 9, 12 are in proportion. The number in the box is 18.
Let us assume the missing number be ‘P’,
Then, P, 24, 9, 12
P: 24 = 9: 12
P/24 = 9/12
9/12 is further simplified by dividing both numerator and denominator by 3.
So, P/24 = 3/4
P = (3 × 24)/4
P = 72/4
P = 18
Therefore, the missing number is 18.

If two ratios are equal, then they are in proportion.
Use Fig. (In which each square is of unit length) for questions 39 and 40:

The ratio of the perimeter of the boundary of the shaded portion to the perimeter of the whole figure is 3: 7.
From the figure, perimeter of shaded portion = 1 + 2 + 1 + 2 = 6 units
Perimeter of whole figure = 3 + 4 + 3 + 4 = 14 units
Then, ratio of the perimeter of the boundary of the shaded portion to the perimeter of the whole figure = 6/14
= 3/7
= 3: 7

The ratio of the area of the shaded portion to that of the whole figure is 1: 6.
Area of the shaded figure = 2 × 1
= 2 sq. units
Area of whole figure = 3 × 4 = 12 sq. units
The ratio of the area of the shaded portion to that of the whole figure is = 2: 12
= 2/12
= 1/6
= 1: 6

The ratio of sleeping time to awaking time is 3: 1.
From the figure, sleeping time = 18 hours
Then, awaking time = 24 – 18 = 6 hours
Therefore, the ratio of sleeping time to awaking time is 18/6
= 3/1
= 3: 1

A ratio expressed in lowest form has no common factor other than one in its terms.

To find the ratio of two quantities, they must be expressed in same units.

Ratio of 5 paise to 25 paise is the same as the ratio of 20 paise to 100 paise.
From the question,
5 paise : 25 paise = 20 paise: [ ]
Let us assume the missing number be Q,
5 paise : 25 paise = 20 paise: Q
5/25 = 20/Q
Q = (20 × 25)/5
Q = 500/5
Q = 100
Therefore, Ratio of 5 paise to 25 paise is the same as the ratio of 20 paise to 100 paise

Saturn and Jupiter take 9 hours 56 minutes and 10 hours 40 minutes, respectively for one spin on their axes. The ratio of the time taken by Saturn and Jupiter in lowest form is 149: 160.
From the question,
Saturn takes 9 hours 56 minutes for one spin on their axes
We know that, 1 hour = 60 minutes
So, (9 × 60) + 56 = 540 + 56 = 596 minutes
Jupiter takes 10 hours 40 minutes for one spin on their axes
= (10 × 60) + 40
= 600 + 40
= 640 minutes
The ratio of the time taken by Saturn and Jupiter in lowest form is = 596/640
Divide both numerator and denominator by 2,
= 298/320
Again, divide both numerator and denominator by 2,
= 149/160
Therefore, the ratio of the time taken by Saturn and Jupiter in lowest form is 149 : 160.

Given, 10 g of caustic soda dissolved in 100 mL water.
The ratio of caustic soda to water should be in proportion.
10g: 100 mL : :x g :1 L
[∵ 1 L = 1000 mL]
x g × 100 mL = 10 g × 1000 mL     [by cross multiplication]

x = 100 g

Given, marked price of a table = ₹ 625
Sale price of a table = ₹ 500
Ratio of sale price to marked price = ₹500/625 = 500/625
= 20/25  [on dividing numerator and denominator by 25]
= 4/5   [on dividing numerator and denominator by 5]
∴ Required ratio  = 4 : 5    

(i) The given pair is equivalent as lowest form of 4/6 is 2/3 and other ratio is 2/3 .
(ii) The given pair is equivalent as lowest form of 8/4 is 2/1 and other ratio is —2/1 
(iii) The given pair is not equivalent as lowest form of 12/20 is 3/5 and other ratio is 4/5 .

Ratio 10:21 is larger as it approximates to 0.476 and 21:93 approximates to 0.225.

Given,
Quantity of Burfi = 18 kg and Khoya : Sugar = 7 : 2
Total of ratio = 7 + 2 = 9
Quantity of Khoya = (18/9) x 7 = 14kg
So, Reshma used 14 kg Khoya.

Given,
Length of the line segment = 56 cm Ratio of two parts = 2 : 5 Sum of ratios = 2 + 5 = 7
Length of first part = (2/7) x 56 = 16 cm
Length of second part = (5/7) x 56 = 40 cm

Number of milk teeth in human beings = 20
Number of permanent teeth in human beings = 32
Ratio of the number of milk teeth to the number of permanent teeth = 20/32
= 5/8 [on dividing numerator and denominator by 4]
= 5 : 8

Sex ratio if there are 3732 females per 4000 males in a town is given as,
= 3732/4000

(a) Ratio of income to income tax is = 360000/24000
=15/1
(b) Ratio of income tax to income after paying income tax
= 24000/336000
= 3/42

(a) Ratio of Ramesh’s earnings to the total earnings of Ramesh and Rama
=28000/64000
=9/16
(b) Ratio of Rama’s earnings to the total earnings of Ramesh and Rama
=36000/64000
= 7/16

No. of people working in a company = 288 
Total no. of men in the company = 112 
Total no. of women in the company = 176
(a) Ratio of the number of men to that of women = 112/176
=7/11
(b) Ratio of the number of women to the total number of persons
= 176/228
=44/57

Given,
Length of rectangular sheet = 1.2 m [v 1 m = 100 cm]
= 1.2 x 100cm= 120cm Width of rectangular sheet = 21 cm
Ratio of width to length = 21 cm/120 cm
= 7/40 = 7:40 [on dividing numerator and denominator by 3]

Speed of scooter =120/3 = 40km/h
Speed of train = 120/2 = 60 km/h
Now, ratio of speed of scooter to speed of train = 40/60 = 2/3

As per given information, total period in office = 9 a.m. to 12 p.m. and 12 p.m. to 5 p.m.
This implies total period = 3 hours + 5 hours  = 8 hours
Since 1 hour = 60 minutes So, 8 hours = 8 × 60 = 480 minutes
Now, Ratio of lunch interval to total period = 30/480= 1/16

Here given, length of shadow of stick = 4m 
Length of stick = 3m 
Also, length of shadow of flagstaff = 24m 
Now, length of flagstaff =(3 x 24)/4 = 18m

As given, 1 cup of milk and 2 1/2 cup of flour to make cake for six persons.
For one person, the ingredients required will be, Milk = 1/6cup
Flour = 5/12 cups
Now, for 8 people, milk 8 x (1/6) = 4/3 cups
Flour = 10/3 cups
Therefore, we need 4/3 cups of milk and 10/3 cups of flour to make cake for eight people.

Given, ratio of number of large classrooms to small classrooms = 3:4 Number of small classrooms = 20 Let the classrooms are multiple of x.
So, large classrooms = 3x Small classrooms = 4x
According to the question, 4x = 20 => x=20/4 = 5 .
Hence, number of large classrooms = 3x
= 3 x 5 = 15

Given, total newspapers = 312 English newspapers = 216
Hindi newspapers = Total number of newspapers – Newspapers in English = 312 – 216 = 96
(a) Ratio of number of English newspapers to number of Hindi newspapers = 216/96
= 9/4 = 9:4 [on dividing numerator and denominator by 24]
(b) Ratio of number of Hindi newspapers to the total number of newspapers = 96/312
= 4/13 = 4:13.

Given, number of Hindu students = 288 Number of Muslim students = 252 Number of Sikh students = 144 Number of Christian students = 72 Total number of students = 288+252+144+72 =756
(a) Ratio of number of Hindu students to the number of Christian students = 288/72
= 4/1 = 4:1 [on dividing numerator and denominator by 72]
(b) Ratio of number of Muslim students to the total number of students = 252/756
= 1/3 =1:3 [on dividing numerator and denominator by 252]

Given, ratio of North Indian food stalls to South Indian food stalls = 5:4
Total number of food stalls =117
Total ratio = 5+4 = 9
North Indian food stalls = (5/9) x 117 = 65
South Indian food stalls = (4/9) x 117 = 52

Given, at parking stand, number of Cycles = 115
Scooters = 75
Bikes = 45
Total number of vehicles = 115+75+ 45 = 235
Ratio of number of cycles to the total number of vehicles = 115/235
= 23/47 =23:47
[on dividing numerator and denominator by 5]

Time taken to travel from Ajmer to Jaipur = 2 hours 
Distance between Ajmer and Jaipur = 130 km 
Also, Distance between Delhi and Bhopal = 780km
Now, speed of train = 130/2 km/h = 65km/h
Thus, time taken by train to travel from Delhi to Bhopal = 780/65 =12 hours

Ratio of length and breadth of a school ground = 150/90 = 5/3
Also, ratio of length and breadth of a mela ground = 210/126 = 5/3
Thus, the given measurements are in proportion.

(a) From given figures, ratio of areas of Africa to Europe =13/5
(b) From given figures, ratio of areas of Australia to Asia = 2/11
(c) From given figures, ratio of areas of Antarctica to Combined area of North America and South America = 13/35

Given, cost of two varities of tea = Rs. 234 and Rs. 130
Ratio of their costs = 234/130 = 9/5 = 9:5
[on dividing numerator and denominator by 26]
Total weight of mixture = 84 kg Total ratio = 9+5 = 14
Weight of first variety tea = (9/14) x 84 = 54 kg
Weight of second variety tea = (5/14) x 84 = 30 kg

Given, the ratio of Zinc and Copper in alloy = 7:9 and weight of alloy = 8 kg
Let the weight of Zinc and Copper in alloy be 7x and 9x respectively, where x is multiple of weight.
Then, total weight =7x+9x = 6x
16x = 8 kg => x = ½ kg
Weight of copper = 9x = 9 x (1/2) = 4 ½ kg Hence, the weight of copper is 4 ½ kg.

(i) AC/AF = 2/5 
(ii) AG/AD = 2/1
(iii) BF/AI = 1/2
(iv) CE/DI = 2/5

Let the two numbers are 9x and 16x, whose sum is 100.
⇒ 9x +16x = 100
⇒ 25x = 100
⇒ x = 4

(i) Ratio of the area of the shaded portion to that of the whole figure
= 8/16
= 1/2
(ii) Ratio of the area of the shaded portion to that of the whole figure
= 16/32
= 1/2

Total pages of manuscript to type = 40
Typed pages of manuscript = 30
Left pages = 40 – 30 = 10
Ratio of the number of pages to the types pages to the number of left pages = 30/10 = 3/1 = 3:1 .

(a) Perimeter of shaded region = 2(20 + 36) = 112cm
Also, perimeter of whole design = 2 (40 + 60) = 200cm
ow, ratio of the perimeter of shaded portion to the perimeter of the whole design = 112/200 = 14/25
(b) Area of shaded portion = 20 x 36 = 720 sq.cm. 
Area of unshaded portion = (40 x 60) - 720 = 1680 sq.cm.
Now, ratio of the area of the shaded portion to the area of the unshaded portion = 720/1680 = 3/7

(a) Ratio of the areas of shaded portion I to shaded portion II is = 5/8
(b) Ratio of the areas of shaded portion II to shaded portion III = 8/7
(c) Ratio of the areas of shaded portions I and II taken together and shaded portion III = 13/7

Distance travelled by car in 15 litres of petrol = 240km
Distance travelled by car in 25 litres of petrol = (240/15) x 25 = 400km

Money earned by Bachhu Manjhi in 8 months = Rs. 24000
(a) Money earned by Bachhu Manjhi in 12 months = (24000/8) x 12 = Rs. 36000
(b) Rs 24000 earned by Bachhu Manjhi in = 8 months
Rs 42000 earned by Bachhu Manjhi in = (8/24000) x 42000 = 16 months

360 quintals yield of wheat is obtained from = 8 hectares
540 quintals yield of wheat is obtained from = (8/360) x 540 = 12 hectares

In 24 hours, Earth rotates by = 360 degrees
In 2 hours. Earth rotates by = (360/24) x 2 = 30 degrees

Tablets required per day = 2 
Tablets required for 15 days = 15 x 2 = 30
Now cost of 10 tablets = Rs. 17
Cost of 30 tablets = (17/10) x30 = 51

Quarterly fee for class VI = Rs 540
Fee for seven months = (540/4) x 7= Rs. 945

As given, votes cast for two of the candidates were in the ratio = 5:7 
Let votes received by the two candidates = 5x and 7x 
Also, votes received by successful candidates = 20734 
This implies, 7x = 20734 x = 2962
Votes received by the opponent = 5 x 2962 = 14810

Weight of 3 metre long pipe = 7.6kg
Weight of 7.8 m long pipe = (7.6/3) x 7.8 = 19.76 kg

5 cups of raspberry juice requires sugar = 5/2 cups
6 cups of raspberry juice requires sugar = 5/(2x5) x 6 = 3 cups

Consider the number of rows = x
This implies, total no. of plants = (number of plant in each row) x (number of rows)
1890 = 63x 
x = 1890/63 = 30
So, no. of rows = 30 
Plants destroyed by worm in each row = 18 
Thus total plants destroyed by worm = 18x30 = 540

Given, length and breadth of the floor of a room = 5m and 3m So area of the floor = 3 x 5 = 15sqm
Also area of one tile = 1/16 sqm
So, area of floor covered by 40 tiles = (1/16) x 40 = 5/2 sqm
Area of floor not covered by tiles = 15 - 5/2 = 25/2 sqm
Now, ratio of the tiled and the non-tiled portion of the floor

Measurement of the board = 3m x 2m 
So, area of the board = 300cm x 200cm = 60000 sqm
Measurement of rectangular piece cut down from board = 250cm × 90cm 
So, area of rectangular piece cut down from board = 22500 sqcm 
Area of remaining portion of the board = 60000 - 22500 = 37500 sqcm
Now, ratio of the area of cut out piece and the remaining piece = 22500/37500.