Ratio & Proportion

The foundation of Arithmetic. Master the art of 'Balancing Ratios', the C&D Rule, and the 'Coins Concept' to solve distribution problems instantly.

1. Balancing Ratios (Zig-Zag Method)

To combine $A:B$ and $B:C$ into $A:B:C$ .

Example:

Q: If

A:B=2:3

and

B:C=5:7

, find

A:B:C

Solution:

A:B = 2:3

B:C = 5:7

.
Multiply top by 5, bottom by 3.

10 : 15 : 21

Basic formulas for $a:b::c:d$ .

Example:

Q: Find Third Proportional to 4 and 6.

Solution:

b^2/a = 6^2/4 = 36/4 = 9

Used when $a/b = c/d$ .

Example:

Q: If

\frac{x+5}{x-5} = \frac{7}{3}

, find x.

Solution: By C&D:

x/5 = (7+3)/(7-3) = 10/4 = 5/2

x = 25/2 = 12.5

Standard Exam Pattern.

Formula: $\text{Value} = \text{Denomination} \times \text{Number of Coins}$ .
Tip: Convert everything to smallest unit (Paise) to avoid decimals.

Example:

Q: Bag has Rs 1, 50p, 25p coins in ratio 2:3:4. Total Rs 180. Find 50p coins.

Solution: Values:

2(100) : 3(50) : 4(25) = 200:150:100 = 4:3:2

.
Sum ratios: 9 units = 18000p. 1 unit = 2000.
Value of 50p =

3 \times 2000 = 6000p

. Coins =

6000/50 = 120

Logic: $\text{Income} - \text{Expenditure} = \text{Savings}$ .

Cross-Product Method: If Ratio of Income is $A:B$ , Exp is $C:D$ , and Savings are $S_1, S_2$ .
$(AD - BC)x = (CS_2 - DS_1)$ . Use this to find 'x' unit value directly.

Example:

Q: Income ratio 3:2, Exp ratio 5:3, Savings Rs 1000 each. Find Income.

Solution: Cross:

(3 \times 3) - (2 \times 5) = 9-10 = -1

.
Savings:

(5 \times 1000) - (3 \times 1000) = 2000

.
1 unit = 2000. Income

A = 3 \times 2000 = 6000