Selina Concise Mathematics class 7 ICSE Solutions – Ratio and Proportion (Including Sharing in a Ratio)
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POINTS TO REMEMBER
- 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. - 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. - 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 = \(\frac { 5 }{ 8 }\) of 64 cm = 40 cm
Second part = \(\frac { 3 }{ 8 }\) 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 = \(\frac { 4 }{ 9 }\) of Rs. 720 = Rs. 320
y’s share =\(\frac { 5 }{ 9 }\) 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 = \(\frac { 3 }{ 12 }\)x 180°= 45°
Second angle = \(\frac { 2 }{ 12 }\) x 180°= 30°
Third angle = \(\frac { 7 }{ 12 }\) 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 \(\frac { 1 }{ 3 }\) : 4 \(\frac { 1 }{ 5 }\) : 6 \(\frac { 1 }{ 8 }\) 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. \(\frac { 810 x 4 }{ 3 }\)
= 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
= \(\frac { 5 }{ 2 }\) x 10=25
and second number = \(\frac { 7 }{ 2 }\) 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 = \(\frac { 168 }{ 21 }\)x 10 =80
Second number = \(\frac { 168 }{ 21 }\)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 = \(\frac { 38 }{ 13 }\) 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 \(\frac { 8 }{ 15 }\) of the whole class.
Question 12.
The population of a town is ’ 50,000, out of which males are \(\frac { 1 }{ 3 }\) 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 = \(\frac { 1 }{ 3 }\) 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
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
= \(\frac { 75 }{ 10 }\) : \(\frac { 25 }{ 10 }\) =3:1
(ii) Ratio between B and total alloy
= 2.5 : 10 = \(\frac { 25 }{ 10 }\) : 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 = \(\frac { 3150 x 10 }{ 21 }\) = 1500 = Rs. 1500
B’s share = \(\frac { 3150 x 11 }{ 21 }\) = 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 = \(\frac { 14400 }{ 16 }\) x 5 = ₹ 4500
B’s share = \(\frac { 14400 }{ 16 }\) x 3 = ₹ 2700
C’s share = \(\frac { 14400 }{ 16 }\) 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 = \(\frac { 168 }{ 28 }\)
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 = \(\frac { 45000 }{ 3}\)
⇒ 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:
