Master Ratio & Proportion from Zero

Ratio is a comparison of two quantities of the same type. If A:B = 3:4, it means for every 3 units of A, there are 4 units of B. Proportion means two ratios are equal: A:B = C:D. Key rule: Product of means = Product of extremes → B×C = A×D.

Key Formulas

Fourth Proportional

If a:b = c:d, then d = bc/a

d is called the fourth proportional.

Mean Proportional

Mean proportional of a and b = √(ab)

When a:x = x:b, then x = √(ab).

Componendo-Dividendo

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)

Very useful trick for advanced problems.

Partnership formula

Profit share ∝ Capital × Time

Used in partnership problems.

⚡ Shortcut Tricks

Splitting a number: To divide N in ratio a:b:c → A gets N×a/(a+b+c), B gets N×b/(a+b+c), C gets N×c/(a+b+c).

Direct proportion: A1/A2 = B1/B2. If one increases, other increases. Example: price ∝ weight.

Inverse proportion: A1×B1 = A2×B2. If one increases, other decreases. Example: speed ∝ 1/time.

Finding A:C from A:B and B:C — multiply ratios: A:C = (A/B) × (B/C). Example: A:B=4:5, B:C=3:4 → A:C=12:20=3:5.

Finding A:D from A:B, B:C, C:D — multiply all three ratios together.

Common factor x: If numbers are in ratio 4:5, assume 4x and 5x. Use given condition to find x.

Coins in bag: Multiply each coin value by its ratio part to get total value per 'x' units.

Adding/removing quantity: Set up equation with new ratio after change. Example: milk:water was 2:5, after adding 4L water ratio became 1:3 → solve for x.

✏️ Solved Examples

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