Ratio & Proportion

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

• Method: If $A:B$ is $2:3$ and $B:C$ is $4:5$.
• Make 'B' common. Multiply first by 4, second by 3.
• Zig-Zag: $(2 \times 4) : (3 \times 4) : (3 \times 5) = 8 : 12 : 15$.

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

• Mean Proportional between $a$ and $b$: $\sqrt{ab}$.
• Third Proportional to $a, b$: $b^2/a$.
• Fourth Proportional to $a, b, c$: $bc/a$.

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

• Rule: $\frac{a+b}{a-b} = \frac{c+d}{c-d}$.
• Shortcut: If you see $\frac{x+a}{x-a} = b$, imply $x/a = (b+1)/(b-1)$.

Standard Exam Pattern.

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

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.