NCERT Solutions Class 6th Maths Chapter – 12 Ratio and Proportion Exercise – 12.1

Question 1. There are 20 girls and 15 boys in the class.

(A) What is the ratio of the number of girls to the number of boys?
(B) What is the ratio of the number of girls to the number of students in the class?

Solution – 

(A) Number of girls = 20
Number of boys = 15
Total number of students = 20 + 15 = 35
∴ Ratio of the number of girls to the number of boys
= Number of girls/Number of boys = 20/15
= 20 ÷ 5/15 ÷ 5 = 4/3 or 4 : 3

Thus, the required ratio is 4 : 3.

(B) Ratio of the number of girls to the number of students
= Number of girls/Number of boys = 20/35
= 20 ÷ 5/35 ÷ 5 = 4/7 or 4 : 7

Thus, the required ratio is 4 : 7.

Question 2. Out of 30 students in a class, 6 like football, 12 like cricket and remaining like tennis. Find the ratio of-

(A) Number of students liking football to the number of students liking tennis.
(B) Number of students liking cricket to total number of students.

Solution –

Number of students in the class = 30
Number of students liking football = 6
Number of students liking cricket = 12
Number of students liking tennis = 30 – (6 + 12) = 30 – 18 = 12

(A) Ratio of the number of the students liking football to the number of students liking tennis

= Number of Student Liking Football/Number of Student Liking Tennis
= 6/12 = 6 ÷ 6/12 ÷ 6 = 1/2 or 1 : 2

Thus, the required ratio is 1 : 2.

(B) Ratio of the number of students liking cricket to the total number of students

= Number of Student Liking Cricket/Total Number of Student
= 12/30 = 12 ÷ 6/30 ÷ 6 = 2/5 or 2 : 5.

Thus, the required ratio is 2 : 5.

Question 3. See the figure and find the ratio of-

(A) Number of triangles to the number of circles inside the rectangle.
(B) Number of squares to all the figures inside the rectangle.
(C) Number of circles to all the figures inside the rectangle.

Solution – 

(A) Number of triangles 3
Number of circles = 2
∴ Ratio of number of triangles to the number of circles
= Number of triangles/Number of circles = 3/2 or 3 : 2

Thus, the required ratio is 3 : 2.

(B) Number of squares = 2
Number of all figures = 7
∴ Ratio of number of squares to the number of all the figures
= Number of Squares/Number of all the figures = 2/7 or 2 : 7

Thus, the required ratio is 2 : 7.

(C) Ratio of number of circles to the number of all the figures
= Number of circles/Number of all the figures = 2/7 or 2 : 7

Thus, the required ratio is 2 : 7.

Solution –

Distance travelled by Hamid = 9 km.
Distance travelled by Akhtar = 12 km.
Speed of Hamid = 9 km
per hour Speed of Akhtar = 12 km per hour

∴ Ratio of the speed of Hamid to the speed of Speed of Hamid ar = Speed of Akhtar
Akhtar = Speed of Hamid/Speed of Akhtar
= 9/12 = 9 ÷ 3/12 ÷ 3 = 3/4 or 3 : 4
Thus, the required ratio is 3 : 4.

Solution –

15 / 18 = (5 × 3) / (6 × 3)
= 5 / 6
5 / 6 = (5 × 2) / (6 × 2)
= 10 / 12
5 / 6 = (5 × 5) / (6 × 5)
= 25 / 30
Hence, 5, 12 and 25 are the numbers which come in the blanks respectively.
Yes, all are equivalent ratios.

Question 6. Find the ratio of the following-

(A) 81 to 108
(B) 98 to 63
(C) 33 km to 121 km
(D) 30 minutes to 45 minutes

Solution –

(a) 81 to 108 81/108 = 81 ÷ 27/108 ÷ 27 = 3/4
[HCF of 81 and 108 = 27
Thus, the ratio = 3 : 4

(b) 98 to 63 = 98/63 = 98 ÷ 7/63 ÷ 7 = 14/9
[HCF of 98 and 63 = 7
Thus, the ratio = 14 : 9

(c) 33 km to 121 km = 33/121 = 33 ÷ 11/121 ÷ 11 = 3/11
[HCF of 33 and 121 = 11
Thus, the ratio = 3 : 11

(d) 30 minutes to 45 minutes
= 30/45 = 30 ÷ 15/45 ÷ 15 = 2/3
[HCF of 30 and 45 = 15
Thus, the ratio = 2 : 3

Question 7. Find the ratio of the following-

(A) 30 minutes to 1.5 hours
(B) 40 cm to 1.5 m
(c) 55 paise to Rs 1
(d) 500 mL to 2 litres

Solution – 

(A) 1 hour = 60 minutes
∴ 1.5 hours = 60 x 1.5 minutes = 90 minutes
∴ Ratio of 30 minutes to 1.5 hours = Ratio of 30 minutes to 90 minutes
= 30/90 = 30 ÷ 30/90 ÷ 30 = 1/3 = 1 : 3
[HCF of 30 and 90 = 30]

(B) 1 m = 100 cm
∴ 1.5 m = 1.5 x 100 cm = 150 cm.
∴ Ratio of 40 cm to 1.5 m = Ratio of 40 cm to 150 cm.
40/150 = 40 ÷ 10/150 ÷ 10 = 4 : 15
[HCF of 40 and 150 = 10]

(C) ₹1 = 100 paise
∴ Ratio of 55 paise to ₹ 1 = Ratio of 55 paise to 100 paise
= 55/100 = 55 ÷ 5/100 ÷ 5 = 11/20 = 11 : 20
[HCF of 55 and 100 = 5]

(D) 500 mL to 2 litres
1 litre = 1000 mL
∴ 2 litres = 2 x 1000 mL = 2000 mL
∴ Ratio of 500 mL to 2 litres = Ratio of 500 mL to 2000 mL
= 500/2000 = 500 ÷ 500/2000 ÷ 500 = 1/4 = 1 : 4
[HCF of 500 and 2000 = 500]

Question 8. In a year, Seema earns ₹ 1,50,000 and saves ₹ 50,000. Find the ratio of-

(A) Money that Seema earns to the money she saves.
(B) Money that she saves to the money she spends.

Solution –

(A) Money earned by Seema = ₹ 1,50,000
Money saved by her = ₹ 50,000
∴ Money spent by her = ₹ 1,50,000 – ₹ 50,000 = ₹ 1,00,000
∴ Ratio of money earned by Seema to the money saved by her
= Money earned/Money saved = 1,50,000/50,000
= 15/5 = 15 ÷ 5/5 ÷ 5 = 3/1 3 : 1

(B) Ratio of money saved by Seema to the money
Spend by her = Money saved/Money spend
= 50,000/1,00,000 = 5/10 = 5 ÷ 5/10 ÷ 5 = 1/2 = 1 : 2

Solution –

Number of teachers = 102
Number of students = 3300
∴ Ratio of number of teachers to the number of students
= Number of teachers/number of students
= 102/3300 = 102 ÷ 6/3300 ÷ 6 = 17/550 = 17 : 550

Question 10. In a college, out of 4320 students, 2300 are girls, find the ratio of-

(A) Number of girls to the total number of students.
(B) Number of boys to the number of girls.
(C) Number of boys to the total number of students.

Solution –

Total number of students = 4320
Number of girls = 2300
∴ Number of boys = 4320 – 2300 = 2020

(A) Ratio of number of girls to the total number of students
= Number of girls/Total number of students
= 2300/4320 = 2300 ÷ 20/4320 ÷ 20 = 115/216 = 115/216

(B) Ratio of number of boys to the number of girls
= Number of boys/Number of girls
= 2020/2300 = 2020 ÷ 20/2300 ÷ 20 = 101/115 = 101 : 115

(C) Ratio of number of boys to the total number of students
= Number of boys/Total number of students
= 2020/4320 = 2020 ÷ 20/4320 ÷ 20
= 101/216 = 101 : 216

Question 11. Out of 1800 students in a school, 750 opted basketball, 800 opted cricket and remaining opted table tennis. If a student can opt only one game, find the ratio of-

(A) Number of students who opted basketball to the number of students who opted table tennis.
(B) Number of students who opted cricket to the number of students opting basketball.
(C) Number of students who opted basketball to the total number of students.

Solution –

Total number of students = 1800
Number of students opting basketball = 750
Number of students who opted cricket = 800
Number of remaining students who opted table tennis = 1800 – (750 + 800)
= 1800 – 1550 = 250

(A) Ratio of number of students opted basketball to the number of students who opted table tennis. Number of students opting basketball Number of students opting table tennis
= Number of students opting basketball/Number of students opting table tennis
= 750/250 = 750 ÷ 250/250 ÷ 250 = 3/1 = 3 : 1

(B) Ratio of the students who opted cricket to the number of students opting basketball
= Number of students opting cricket/Number of students opting basketball
= 800/750 = 800 ÷ 50/750 ÷ 50 = 16/15 = 16 : 15

(C) Ratio of number of students who opted basketball to the total number of students
= Number of students who opted basketball/Total number of students
= 750/1800 = 750 ÷ 150/1800 ÷ 150 = 5/12 = 5 : 12

Question 12. Cost of a dozen pens is ₹180 and cost of 8 ball pens is ₹56. Find the ratio of the cost of a pen to the cost of a ball pen.

Solution –

Cost of 1 dozen, i.e., 12 pens = ₹180
∴ Cost of 1 pen = ₹180 / 12 = ₹15
Cost of 8 ball pens = ₹56

∴ Cost of 1 ball pen = ₹ 56 / 8 = ₹ 7
Ratio of cost of 1 pen to cost of 1 ball pen
= Cost of 1 pen/Cost of 1 ball pen = 15/7 = 15 : 7

Thus required ratio is 15 : 7.

Question 13. Consider the statement – Ratio of breadth and length of a hall is 2 : 5. Complete the following table that shows some possible breadths and lengths of the hall.

Solution –

Solution –

We have 3 + 2 = 5
Total number of pen = 20
∴ Sheela’s share = 3 / 5 x 20 = 3 x 4 = 12 pens 
Sangeeta’s shares = 2 / 5 x 20 = 2 x 4 = 8 pens.
Thus Sheela gets 12 pens and Sangeeta gets 8 pens.

Solution –

Given that, Money got by Shreya – Money got by Bhoomika = 15 : 12
∴ Sum = 15 + 12 = 27Share of Shreya = 15×36/27 = ₹ 20
Share of Bhoomika = 12×36/27 = ₹ 16

Question 16. Present age of father is 42 years and that of his son is 14 years. Find the ratio of

(A) Present age of father to the present age of son.
(B) Age of the father to the age of son, when son was 12 years old.
(C) Age of father after 10 years to the age of son after 10 years.
(D) Age of father to the age of son when father was 30 years old.

Solution –

Present age of father = 42 years.
Present age of his son = 14 years.
(A) Ratio of present age of father to the present age of son
= 42/14 = 42/14 / 14/14 = 3 : 1

(B) When son was 12 years old, i.e., 14 – 12 = 2 years ago father’s age = 42 – 2 = 40 years. Ratio of the father’s age to the son’s age
= 40/12 = 40/4 / 12/4 = 10/3 = 10 : 3

(C) Ratio of father’s age after 10 years, i.e., 42 + 10 = 52 years
to the age of son after 10 years, i.e., = 14 + 10 = 24 years
= 52/24 = 52/4 / 24/2 = 13/6 = 13 : 6

(D) Ratio of the son’s age to the age of father when he was only 30 years .
When father was 30 years,
i.e., before 42 – 30 = 12 years
Age of son was = 14 – 12 = 2 years
∴ Required ratio
= 30/2 = 30/2 / 2/2 = 15/1 = 15 : 1