Selina Concise Solutions for Chapter 6 Ratio and Proportion (including Sharing in Ratio) Class 7 ICSE Mathematics

Exercise 6 (A)

1. Express each of the given ratio in its simplest form :
(i) 22 : 66
(ii) 1.5 : 2.5

(iv) 40 kg : 1 quintal
(v) 10 paise : ₹ 1
(vi) 200m : 5 km
(vii) 3 hours : 1 day
(viii) 6 months : 4/3 years

Answer

2. Divide 64 cm long string into two parts in the ratio 5 : 3 .

Answer

Sum of ratios = 5+3 = 8
∴ first part = 5/8 of 64 cm = 40 cm
Second part = 3/8 of 64 cm = 24 cm

3. Rs. 720 is divided between x and y in the ratio 4 : 5 . How many rupees will each get ?

Answer

Total amount = Rs. 720 Ratio between x, y = 4 : 5
Sum of ratios = 4+ 5 = 9
x's share = 4/9 of Rs. 720 = Rs. 320
y's share = 5/9 of Rs. 720 = Rs. 400

4. The angles of a triangle are in the ratio 3 : 2 : 7. Find each angle.

Answer

Ratio in angles of a triangle = 3 : 2 : 7
Sum of ratios = 3 + 2 + 7 = 12
Sum of angles of a triangle = 180°
∴ First angle = 3/12 ×180°= 45°
Second angle = 2/12 ×180°= 30°
Third angle = 7/12 ×180°= 105°

5. A rectangular field is 100 m by 80 m. Find the ratio of

(i) length to its breadth

(ii) breadth to its perimeter.

Answer

Length of field (l) = 100 m

Breadth (b) = 80 m

∴ Perimeter = 2 (l + b) = 2 (100 + 80) m = 2×180 = 360 m

(i) Ratio between length and breadth

= 100 : 80 = 5 : 4

(Dividing by 20, the HCF of 100 and 80)

(ii) Ratio between breadth and its perimeter

= 80 : 360 = 2 : 9

(Dividing by 40, the HCF of 80 and 360)

6. The sum of three numbers, whose ratios are is 4917.Find the numbers.

Answer

Sum of three numbers = 4917
Ratio between them =

7. The ratio between two quantities is 3 : the first is Rs. 810, find the second.

Answer

Ratio between two quantities = 3 : 4
Sum of ratio = 3+ 4 = 7
∴Second quantity = Rs. (810 × 4)/3
= Rs. 270 × 4 = Rs. 1080

8. Two numbers are in the ratio 5 : 7. Their difference is 10. Find the numbers .

Answer

Ratio between two numbers = 5:7

Difference = 7-5 = 2

If difference is 2, then first number = 5

and if difference is 10, then first number

= 5/2 × 10=25

and second number = 7/2 × 10 = 35

9. Two numbers are in the ratio 10 : 11. Their sum is 168. Find the numbers.

Answer

Ratio between two numbers = 10 : 11

Sum of ratios = 10 + 11=21

Total sum = 168

∴ first number = 168/21 × 10 = 80

Second number = 168/21 × 11 = 88

10. A line is divided in two parts in the ratio 2.5 : 1.3. If the smaller one is 35T cm, find the length of the line.

Answer

Ratio between two parts of a line

= 2-5 : 1-3 =25 : 13

Sum of ratios = 25 + 13 = 38

Length of smaller part = 35.1 cm 38

Now length of line = 38/13 × 35.1 cm

= 38 × 2.7 cm

= 102.6 cm

11. In a class, the ratio of boys to the girls is 7:8. What part of the whole class are

girls.

Answer

Ratio between boys and girls = 7:8

Sum of ratios = 7 + 8 = 15

∴ Girls are 8/15 of the whole class.

12. The population of a town is ’ 50,000, out of which males are 1/3 of the whole population. Find the number of females. Also, find the ratio of the number of females to the whole population.

Answer

Total population = 180,000

Population of males = 1/3 of 180,000 = 60,000

∴ Population of females = 180,000 – 60,000 = 120,000

Ratio of females to whole population

= 120,000 : 180,000 = 2:3

13. Ten gram of an alloy of metals A and B contains 7.5 gm of metal A and the rest is metal B. Find the ratio between :

(i) the weights of metals A and B in the alloy.

(ii) the weight of metal B and the weight of the alloy.

Answer

Total weight of A and B metals = 10 gm A’s weight = 7.5 gm B’s weight = 10 – 7.5 = 2.5

(i) Ratio between A and B = 7.5 : 2.5
= 75/10 : 25/10 = 3 : 1
(ii) Ratio between B and total alloy
= 2.5 : 10 = 25/10 : 10
= 25 : 100 = 1 : 4

14. The ages of two boys A and B are 6 years 8 months and 7 years 4 months respectively. Divide Rs. 3,150 in the ratio of their ages.

Answer

A’s age = 6 years 8 months

= 6×12 + 8 = 72 + 8 = 80 months

B’s age = 7 years 4 months = 7×12 + 4 = 84 + 4 = 88 months

∴ Ratio between them = 80 : 88 = 10 : 11

Amount = Rs. 3150

Sum of ratios = 10 + 11 =21

∴ A’s share = (3150×10)/21 = 1500 = Rs. 1500

B’s share = (3150×10)/21 = 1650 = Rs. 1650

15. Three persons start a business and spend Rs. 25,000; Rs. 15,000 atid Rs. 40,000 respectively. Find the share of each out of a profit of Rs. 14,400 in a year.

Answer

A’s investment = Rs. 25000

B’s investment = Rs. 15000

C’s investment = Rs. 40000

∴ Ratio between their investment

= 25000 : 15000 : 40000

= 5 : 3 : 8

Sum of ratios = 5 + 3 + 8=16 Total profit = ₹ 14400

∴ A’s share = (14400/16) × 5 = ₹ 4500

B’s share = (14400/16) × 3 = ₹ 2700

C’s share = (14400/16) × 8 = ₹ 7200

16. A plot of land, 600 sq m in area, is divided between two persons such that the first person gets three-fifth of what the second gets. Find the share of each.

Answer

Area of plot of land = 600 sq. meter
Let second's share = x
Then first share = 3x/5
∴ Ratio between them
3x/5 : x
⇒ 3/5 : 1 = 3 : 5
Sum of ratios = 3 + 5 = 8
∴ Share of first person = (600/8) × 3
= 225 sq. m
and second share = (600/8) × 5 = 375 sq.m

17. Two poles of different heights are standing vertically on a horizontal field. At a particular time, the ratio between the lengths of their shadows is 2 :3. If the height of the smaller pole is 7.5 m, find the height of the other pole.

Answer

Ratio between the shadows of two poles = 2 : 3
∴ Height of smaller pole = 7.5 m
Height of taller pole = (7.5 × 3)/2
= 22.5/2 = 11.25 m

18. Two numbers are in the ratio 4 : 7. If their L.C.M. is 168, find the numbers.

Answer

Given, Ratio in two numbers = 4:7

and their L.C.M. = 168

Let first number = 4x

and second number = 7x

Now, L.C.M. of 4x and 7x

= 4 × 7 × x = 28x

∴ 28x = 168

x = 168/28

x = 6

∴ Required numbers = 4x and 7x = 4 × 6 = 24 and 7 × 6 = 42

19. is divided between A and B in such a way that A gets half of B. Find :

(i) the ratio between the shares of A and B.

(ii) the share of A and the share of B.

Answer

Total amount to be divided between A and B = ₹ 300
(i) A gets half of B
Hence, ratio between A and B = 1/2
= 1 : 2
(ii) Sum of ratios = 1 + 2 =3
∴ A' shares = (300 × 1)/3 = ₹ 100
∴ B' shares = (300 × 2)/3 = ₹ 200

20. The ratio between two numbers is 5 : 9. Find the numbers, if their H.C.F. is 16.

Answer

Let the first number be 5x and second number be 9x

H.C.F. of 5x and 9x = Largest number common to 5x and 9x = x

Given H.C.F. = 16

⇒ x = 16

∴ Required numbers = 5x and 9x = 5×16 and 9×16 = 80 and 144

21. A bag contains ₹ 1,600 in the form of ₹10 and ₹20 notes. If the ratio between the numbers of ₹10 and ₹20 notes is 2 : 3; find the total number of notes in all.

Answer

Total amount in the bag = 1600
It contains notes in the denomination of ₹ 10 and 20
Ratio between the number of ₹10 and 20
notes is = 2 : 3
Let number of ₹10 note = x
and number of ₹20 notes = y
According to condition,
10x + 20y = 1600 ....(i)
and x = 2y/3 ...(ii)
Now, substitute the value of x in eq. (i)

∴ y = 60
Now, substitute the value of y in eq. (ii), we get
x = (2/3) × 60 = 40
Total number of notes in all = x + y
= 60 +40 = 100 notes

22. The ratio between the prices of a scooter and a refrigerator is 4 : 1. If the scooter costs ₹45,000 more than the refrigerator, find the price of the refrigerator.

Answer

Ratio between the prices of scooter and a refrigerator = 4:1

Cost price of scooter = ₹45,000

Let the cost of scooter = 4x

Cost of refrigerator = 1x

According to condition,

Cost of scooter > Cost of refrigerator

⇒ 4x - 1x = 45000

⇒ 3x = 45000

x = 45000/3

⇒ x = ₹15000

∴ Price of refrigerator = ₹15000

Exercise 6 (B)

1. Check whether the following quantities form a proportion or not ?
(i) 3x , 7x, 24 and 56
(ii) 0.8, 3, 2.4 and 9

(iv) 0.4, 0.5, 2.9 and 3.5

Answer

(i) 3x , 7x, 24 and 56.
If these are in proportion, then
3x × 56 = 7x × 24
⇒ 168x = 168x
which is true.
Hence 3x, 7x, 24 and 56 are in proportion.

(ii) 0.8, 3, 2.4 and 9 are in proportion
if 0.8 × 9 = 3× 2.4
⇒ 7.2 = 7.2
which is true
Hence 0.8, 3, 2.4 and 9 are in proportion.

(iv) 0.4, 0.5, 2.9 and 3.5 are in proportion
if 0.4 × 3.5 = 0.5 × 2.9
⇒ 1.40 = 1.45
which is not true
Hence, 0.4, 0.5, 2.9 and 3.5 are not in proportion.

2. Find the fourth proportional of
(i) 3, 12 and 4
(ii) 5, 9 and 45
(iii) 2.1, 1.5 and 8.4
(iv) 1/3, 2/5 and 8.4
(v) 4 hours 40 minutes, 1 hour 10 minutes and 16 hours.

Answer

(i) 4 th proportional to 3, 12 and 4
= (12×4)/3

= 16

(ii) Fourth proportional to 5, 9 and 45
= (9×45)/5

= 81

(iii) Fourth proportional to 2.1, 1.5 and 8.4
= (1.5×8.4)/2.1

= 1.5 × 4

= 6.0

(iv) Fourth proportional to 1/3, 2/5 and 8.4

(v) Fourth proportional to 4 hours 40 minutes,
1 hours 10 minutes and 16 hours
4 hours 40 minutes = 4 × 60 + 40
= 240 + 40 = 280
1 hour 10 minutes = 1 × 60 + 10
= 60 + 10 = 70 minutes
16 hours = 16 × 60 = 960 minutes
∴ Fourth proportional = (70 × 960)/280
= 240 minutes = 240/60 = 4 hours

3. Find the third proportional of
(i) 27 and 8
(ii) 2 m 40cm and 40 cm
(iii) 1.8 and 0.6
(iv) 1/7 and 3/14
(v) 1.6 and 0.8

Answer

(i) Third proportional to 27 and 9
= (9 × 9)/27

= 3

(ii) Third proportional to 2 m 40 cm and 40 cm
or 240 cm and 40cm
= (40×40)/240

= 20/3

(iii) Third proportional to 1.8 and 0.6
= (0.6×0.6)/1.8

= 0.36/1.8

= 36/180
= 1/5

= 0.2

4. Find the mean proportional between
(i) 16 and 4
(ii) 3 and 27
(iii) 0.9 and 2.5
(iv) 0.6 and 9.6
(v) 1/4 and 1/16

Answer

5. (i) If A : B = 3 : 5 and B : C = 5 : 7, find A : B : C
(ii) If x : y = 2 : 3 and y : z = 5 : 7, find x : y : z
(iii) If m : n = 4: 9 and n : s = 3 : 7, find m : s
(iv) If P : Q = 1/2 : 1/3 and find P : R.
(v) If a: b = 1.5 : 3.5 and b : c= 5 : 6, find a : c.
Find P : r