Selina Concise Mathematics class 7 ICSE Solutions Chapter 6 – Ratio and Proportion (Including Sharing in a Ratio)

Selina Concise Mathematics class 7 ICSE Solutions – Ratio and Proportion (Including Sharing in a Ratio)

Selina Concise Mathematics class 7 ICSE Solutions – Ratio and Proportion (Including Sharing in a Ratio)

POINTS TO REMEMBER

1. Ratio

A ratio is a method to compare two quantities of the same kind with same unit; by dividing the first quantity by the second. The symbol (:) is used for ratio between two quantities e.g. a : b.

Note:

(i) A ratio is a pure number and has no unit.

(ii) A ratio must always be expressed in its lowest terms in simplest form.

(iii) If each term of a ratio is multiplied or divided by the same number or quantity, the ratio remains the same.

2. Proportion :

Proportion is equality of two ratios : e.g. a : b = c : d

i.e. Ratio between first and second is equal to ratio between third and fourth term.

(ii) a and d are called extreme terms and b and c are called mean terms

and a x d = b x c
(iii) Fourth term is called fourth proportional.
3. Continued Proportion

Three quantities are called in continued proportion if the ratio between first and second is equal to the ratio between second and third i. e.

a, b, c are in continued proportion if a : b = b : c

b the middle term is called the mean proportional between a and c and c, the third term is called the third proportional to a and b.

EXERCISE 6 (A)

Question 1.
Express each of the given ratio in its simplest form :

Answer:

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

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

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

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

Question 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 =  x 180°= 45°
Second angle =  x 180°= 30°
Third angle = x 180°= 105°

Question 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 x 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)

Question 6.
The sum of three numbers, whose ratios are 3  : 4  : 6  is 4917.Find the numbers.

Answer:

Question 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. = Rs. 270 x 4 = Rs. 1080

Question 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
=  x 10=25
and second number =  x 10 = 35

Question 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 =  x 10 =80
Second number =   x 11 =88 Ans.

Question 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 =  x 35.1 cm
= 38 x 2.7 cm = 102.6 cm

Question 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 of the whole class.

Question 12.
The population of a town is ’ 50,000, out of which males are 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 =   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

Question 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 gm

(i) Ratio between A and B = 7.5 : 2.5

=  :  =3:1

(ii) Ratio between B and total alloy
= 2.5 : 10 =   : 10

⇒ 25 : 100 = 1 : 4

Question 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 x 12 + 8 = 72 + 8 = 80 months
B’s age = 7 years 4 months = 7 x 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 = = 1500 = Rs. 1500
B’s share = = 1650 = Rs. 1650

Question 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 =  x 5 = ₹ 4500
B’s share =   x 3 = ₹ 2700

C’s share =   x 8 = ₹ 7200

Question 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:



Question 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:


Question 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 x 7 x x = 28x
∴ 28x = 168
x = 
x = 6
∴ Required numbers = 4x and 7x = 4 x 6 = 24 and 7 x 6 = 42

Question 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:


Question 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

Question 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:


Question 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 = 
⇒ x = ₹15000
.’. Price of refrigerator = ₹15000

EXERCISE 6 (B)

Question 1.
Check whether the following quantities form a proportion or not ?

Answer:

Question 2.
Find the fourth proportional of

Answer:

Question 3.
Find the third proportional of

Answer:

Question 4.
Find the mean proportional between

Answer:

Question 5.


Answer:

Question 6.
If x: y – 5 :4 and 2 : x = 3 :8, find the value of y.
Answer:

Question 7.
Find the value of x, when 2.5 : 4 = x : 7.5.
Answer:

Question 8.
Show that 2, 12 and 72 are in continued proportion.
Answer: