## 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 = 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:**

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