## Ratio and Proportion & Mixtures

Theory

“Ratio and Proportion" is one of the easiest topics in quantitative section in IBPS. It is for this reason that questions from these topics are often asked in conjunction with other topics. The fundamentals of this topic are hence important not just from a stand-alone perspective, but also to answer questions from other topics.

Theory

Properties of Ratios
● A ratio need not be positive. However, if we are dealing with quantities of items, their ratios will be positive
● A ratio remains the same if both numerator and denominator are multiplied or divided by the same non-zero number, i.e.,
● $$\frac{a}{b} = \frac{pa}{pb} = \frac{qa}{qb} , p,q \neq 0$$
● $$\frac{a}{b} = \frac{a/p}{b/p} = \frac{a/q}{b/q} , p,q \neq 0$$

Theory

Properties of Proportions
If a:b = c:d is a proportion, then
● b:a = d:c
● c:d = a:b
● c:a = d:b
● Product of extremes = product of means i.e., ad = bc
● Denominator addition/subtraction: a:a+b = c:c+d , a:a-b = c:c-d
● a,b,c,d,.... are in continued proportion means, a:b = b:c = c:d = ....

Tip

If a, b, c, d and x are positive integers such that $$\frac{a}{b}=\frac{c}{d}$$

1. $$\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$$

2. $$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$

Shortcuts

Alligation Rule: This is used to find the ratio of individual components in a mixture. If two components A and B, costing Rs. X and Rs. Y individually, are mixed and the resultant mixture has an average price of Rs. Z, then the ratio of A and B in the mixture is $$\frac{Z-Y}{X-Z}$$

Solved Example

A Father and his son’s current age ratio is 7:3. After 6 years father’s age will be twice of his son’s age. What is the current age of the father?

a: 42
b: 36
c: 30
d: 45

Explanation:
Let’s say current age is 7x and 3x
After 6 years, (7x+6):(3x+6)= 2:1
Now after solving above equation, we will get x=6 and Father’s age will be 42.

Solved Example

A class has ratio of girls and boys as 3:4. 30 girls are added in the class and new ratio will become 2:1. What are the total number of girls in the class?

a: 42
b: 46
c: 49
d: 48

Explanation:
Let’s girls and boys are 3x and 4x. After adding 30 girls new ratio will be (3x+30):4x=2:1
After solving above equation, we will get x=6.