**Ratio and Proportion: **Ratio and Proportion is one of the important scoring topics of Quantitative Aptitude. There are many exams going to be held in the upcoming months like SSC CGL, SSC CHSL etc., candidates must have started preparing thoroughly for the difficult sections. Quantitative ability questions are asked in almost every government job exam with special focus on ratio and proportion. Here we are providing you quick revision notes on Ratio and Proportion including definitions, formulas, tips and tricks to solve questions.

## ratio and proportion

Ratio and proportion are fundamentally different. When the fraction **a : b** is represented as a ratio and a **Proportion states that two ratios are equal.** a and b are any two integers. Ratio and proportion are two important concepts, and it is the basis of understanding various concepts in mathematics.

### What is the ratio?

Ratio can be defined as a relationship between two quantities such that a : b, where b is not equal to 0. Two numbers can be compared in a ratio only if they are in the same units. Ratio is used to compare two things. **Ratio is represented by ':' symbol**, The ratio can be represented as follows.

- a to b
- a : b
- a/b

**For example,** The ratio of 4 to 8 is represented as 4:8 = 1:2. And the statement is stated in proportion

### What is proportion?

Proportion is an equation that defines how two given ratios are equal to each other. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other. There are 3 types of proportions i.e.

- direct proportion
- inverse proportion
- respectively proportion

## Ratio and Proportion Formula

**ratio and proportion**

a : b ∷ c : d

Product of middle terms = Product of terminal terms

**a****d = b****c**

**fourth proportional**

a : b ∷ c : x

x → fourth proportional

**x=(b×c)/a**

**Ratio and Proportion Questions:**

Example.Find the fourth ratio of the numbers 4,10,12.

Sol. fourth proportional

=(12×10)/4

= 30

**Third proportional →**

a : b ∷ b : x

x → third proportional

**Third proportional of a, b = b²/a**

Now let's look at some more ratio and proportion questions

Find the third proportional of the numbers 4, 12.

Sol. **third**** proportionate**

=(12×12)/4

= 36

**mean proportional**

a : x ∷ x : b

x → **mean proportional **

**of ab mean proportional is = √ab**

## Ratio and Proportion Tricks

Let us learn some rules and tricks to solve questions based on Ratio and Proportion topic here.

- If u/v = x/y, then uy = vx
- If u/v = x/y, then u/x = v/y
- If u/v = x/y, then v/u = y/x
- If u/v = x/y, then (u+v)/v = (x+y)/y
- If u/v = x/y, then (uv)/v = (xy)/y
- If u/v = x/y, then (u+v)/ (uv) = (x+y)/(xy), (Componento-dividend rule)
- If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c.

## Examples of ratio and proportion:

4, 16'smean proportionalFind out?

Sol. **mean proportional **= √(4×16)

=√64

= 8

- If two numbers are in the ratio a : b and their sum is x, then these numbers will be-

**ax/(a+b) & bx/(a+b)**

**Example- If A : B = 3 : 5 and B : C = 9 : 10, then A : B : C**

Sol. A : B = 3 : 5

B : C = 9 : 10

A : B : C = 3 × 9 : 9 × 5 : 5 × 10

= 27 : 45 : 50

- If a : b = n₁ : d₁ , b : c = n₂ : d₂ , c : d = n₃ : d₃

**a : b : c : d = n₁ × n₂ × n₃ : d₁ × n₂ × n₃ : d₁ × d₂ × n₃ : d₁ × d₂ × d₃**

**Example- If A : B = 2 : 3, B : C = 4 : 5, C : D = 6 : 7. Then A : B : C : D.**

Sol. A : B : C : D = 2 × 4 × 6 : 3 × 4 × 6 : 3 × 5 × 6 : 3 × 5 × 7

= 48 : 72 : 90 : 105

= 16 : 24 : 30 : 35

- If two numbers are in the ratio a : b and x is added to both the numbers, then the ratio becomes c : d. So two numbers are given

**ax(cd)/(ad-bc) & bx(cd)/(ad-bc)**

**Example. ****If two numbers are in the ratio 3 : 4. If 8 is added to both the numbers, the ratio becomes 5 : 6. Find the numbers.**

Sol. 1st number

=(3×8 (5-6))/(3×8-5×4)

=(24(-1))/(-2)=12

Number=(4×8 (5-6))/(3×6-5×4)

=(32×(-1))/((-2))=16

- If two numbers are in the ratio a : b, then the number that should be added to each number so that the ratio becomes c : d is

**(ad-bc)/(cd)**

**Example. Find the number which should be added to the ratio 11 : 29 to make it equal to the ratio 11 : 20.**

Sol. Number =(ad – bc)/(c – d)

=(11×20-29×11)/(11-20)=11

- The ratio of income of two persons is → a : b and the ratio of their expenditure is → c : d. If each person's savings is S, then his income is

**aS(dc)/(ad-bc) & bS(dc)/(ad-bc)**

and their expenditure is given

**cS(ba)/(ad-bc) & dS(ba)/(ad-bc)**

**Example. The annual salaries of A and B are in the ratio 5 : 4 and their annual expenses are in the ratio 4 : 3. If each person saves Rs 800 at the end of the year, find their income.**

Sol. A's Incomes

=(5×500(3-4))/(15-16)b

= Rs 2500.

B's Income

=(4×500(3-4))/(15-16)=2000 Rs.

- When two materials A and B in quantities q₁ & q₂, whose cost price per unit are c₁ & c₂ respectively, are mixed to get a mixture c, whose cost price is cm/cm, then

(a) In what ratio are A & B mixed?

**(q₁)/q₂ =(c₂-cm)/(cm-c₁ )**(b) Purchase price of the mixture

**cm = (c₁×q₁+c₂×q₂)/(q₁+q₂)****Example. In what ratio should two types of tea be mixed, one at Rs 9/kg and the other at Rs 15/kg, so that the price of the mixture becomes Rs 10.2/kg?**Sol.

q₁/q₂ =(15-10.2)/(10.2-9)=4.8/1.2

= 4 : 1

**Example. In a mixture of two types of oil O₁ & O₂, the ratio of O₁ : O₂ is 3 : 2, if the price of oil O₁ is Rs. 4/litre. and price of oil O₂ is Rs. 9/litre, then find the purchasing price of the resulting mixture.**Sol.

c_m=(c₁×q₁+c₂×q₂)/(q₁+q₂ )

=(4×3+9×2)/(3+2)

=(12+18)/5=30/5=Rs.6

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