Ratio, Proportion and Rates of Change

This lesson deepens your understanding of ratio and proportion, introducing algebraic expressions for direct and inverse proportion, exponential growth and decay, and rates of change in context.

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Ratio — Advanced Problems

Combining Ratios

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C. Make B the same in both: A : B = 8 : 12, B : C = 12 : 15 → A : B : C = 8 : 12 : 15

Ratio and Algebraic Equations

Example: A and B share £720 in the ratio (x + 1) : (2x − 1). A receives £270. Find x. A's fraction = (x + 1)/(3x) = 270/720 = 3/8 8(x + 1) = 3(3x) → 8x + 8 = 9x → x = 8

Proportion in Geometry

In similar shapes:

• The ratio of areas = (ratio of lengths)²
• The ratio of volumes = (ratio of lengths)³

Example: Two similar cones have heights in the ratio 2 : 3.

• Length ratio = 2 : 3
• Area ratio = 2² : 3² = 4 : 9
• Volume ratio = 2³ : 3³ = 8 : 27

If the smaller cone has a surface area of 80 cm², find the surface area of the larger cone: 80/A = 4/9 → A = 80 × 9/4 = 180 cm²

If the smaller cone has a volume of 64 cm³, find the volume of the larger cone: 64/V = 8/27 → V = 64 × 27/8 = 216 cm³

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Direct and Inverse Proportion

Algebraic Form

Direct proportion: y = kx (graph: straight line through origin) y ∝ x²: y = kx² (graph: parabola through origin) y ∝ √x: y = k√x (graph: half of a parabola sideways)

Inverse proportion: y = k/x (graph: hyperbola) y ∝ 1/x²: y = k/x²

Finding k: substitute the given values, solve for k, then use the formula.

Example (y ∝ x²): y is directly proportional to x². When x = 3, y = 36. Find y when x = 5.

Step 1 — Write the formula: y = kx² Step 2 — Find k: 36 = k × 3² = 9k → k = 4 Step 3 — Use the formula: y = 4 × 5² = 4 × 25 = 100

Example: y is inversely proportional to x². When x = 2, y = 9. Find y when x = 6. y = k/x²; 9 = k/4 → k = 36; y = 36/36 = 1

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Percentage Growth and Decay

Repeated Proportional Change (Compound)

After n applications of multiplier m: Final value = Initial value × mⁿ

Compound interest: A = P(1 + r/100)ⁿ P = principal, r = rate per period (%), n = number of periods.

Example: £3,500 at 4.5% p.a. for 6 years: A = 3500 × 1.045⁶ = 3500 × 1.30226 = £4,557.91

Compound depreciation: A = P(1 − r/100)ⁿ