Formula Sheet • Ratios, Proportions & Compound Rates

Ratio & Proportion Formula Sheet

Essential formulas for simplifying and dividing ratios, unit ratios 1:n, geometric scale factors for area and volume, direct and inverse proportion models, and compound measures.

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Ratios, Sharing & Scale Factors

Simplifying a Ratio

a : b = \frac{a}{\gcd(a, b)} : \frac{b}{\gcd(a, b)}

Divide all parts of the ratio by their greatest common factor (HCF / GCD).

Worked Example

\displaystyle 12 : 18 \implies \gcd=6 \implies \frac{12}{6} : \frac{18}{6} = 2 : 3

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Unit Ratio (1 : n or n : 1)

a : b = 1 : \frac{b}{a} \quad \text{or} \quad \frac{a}{b} : 1

Divide both terms by a to achieve form 1 : n; useful for maps and scale rates.

Worked Example

\displaystyle 4 : 14 \implies 1 : \frac{14}{4} = 1 : 3.5

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Sharing an Amount in Ratio

\text{1 Part} = \frac{\text{Total}}{\sum \text{parts}}, \quad \text{Share} = \text{parts} \times \text{1 Part}

Sum the parts first to determine the value of a single unit share.

Worked Example

\displaystyle 60 \text{ in } 2:3 \implies 2+3=5 \implies \frac{60}{5}=12 \implies 24 : 36

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Find Total from Known Share

\text{Total} = \frac{\text{Known Share}}{\text{Specific Parts}} \times \sum \text{parts}

Find 1 part by dividing the known share by its ratio parts, then multiply by sum.

Worked Example

\displaystyle \text{Ratio } 3:5, \text{ smaller has } 21 \implies 1\text{ pt}=\frac{21}{3}=7 \implies \text{Tot}=7 \times 8 = 56

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Ratio as Fractions

a : b \implies \text{Fraction of total} = \frac{a}{a+b}

Converts part-to-part ratio comparisons into part-to-whole fractions.

Worked Example

\displaystyle 3 : 7 \implies \frac{3}{3+7} = \frac{3}{10} = 30\% \text{ of whole}

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Similar Shapes Scale Factors

\text{Length} = k, \quad \text{Area} = k^2, \quad \text{Volume} = k^3

k = linear scale factor • Areas scale by k² and volumes scale by k³.

Worked Example

\displaystyle k = 3 \implies \text{Area SF} = 3^2 = 9, \quad \text{Vol SF} = 3^3 = 27

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Direct Proportion

Direct Linear Proportion

y = kx \iff \frac{y}{x} = k

k = constant of proportionality • Graph is a straight line passing through (0, 0).

Worked Example

\displaystyle y=15 \text{ when } x=3 \implies k=\frac{15}{3}=5 \implies y=5x

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Direct Proportion to Powers

y = kx^2 \quad \text{or} \quad y = k\sqrt{x}

Frequently models physical phenomena like kinetic energy (E ∝ v²).

Worked Example

\displaystyle y=36 \text{ when } x=3 \implies 36=k(3^2) \implies k=4 \implies y=4x^2

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Unitary Method (Rate Scaling)

y_2 = \left(\frac{y_1}{x_1}\right) \times x_2

Calculate the unit cost/rate first, then scale up to the desired target quantity.

Worked Example

\displaystyle 4 \text{ books cost } 24 \implies 1\text{ bk}=6 \implies 7\text{ bks}=6 \times 7 = 42

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Inverse Proportion

Inverse Proportion Model

y = \frac{k}{x} \iff xy = k

As x doubles, y halves; the product xy remains strictly constant.

Worked Example

\displaystyle x=4, y=6 \implies k = 4 \times 6 = 24 \implies y = \frac{24}{x}

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Inverse Square Law

y = \frac{k}{x^2}

Models light intensity, gravitational force, and electromagnetic radiation drop-off.

Worked Example

\displaystyle y=5 \text{ when } x=2 \implies k = 5(2^2) = 20 \implies y = \frac{20}{x^2}

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Constant Product Rule

x_1 y_1 = x_2 y_2 \implies y_2 = \frac{x_1 y_1}{x_2}

Used directly for worker allocation, speed versus time, and gear ratio problems.

Worked Example

\displaystyle 3 \text{ workers take } 8\text{ hrs} \implies 6 \text{ workers take } \frac{3 \times 8}{6} = 4\text{ hrs}

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Compound Measures & Rates

Speed, Distance & Time

s = \frac{d}{t}, \quad d = s \times t, \quad t = \frac{d}{s}

s = speed • d = distance • t = time. Keep units consistent (km/h vs m/s).

Worked Example

\displaystyle d = 120\text{ miles}, \; t = 2.5\text{ hrs} \implies s = \frac{120}{2.5} = 48\text{ mph}

Density, Mass & Volume

\rho = \frac{m}{V}, \quad m = \rho \times V, \quad V = \frac{m}{\rho}

ρ = density • m = mass • V = volume (e.g., g/cm³ or kg/m³).

Worked Example

\displaystyle m = 54\text{ g}, \; V = 20\text{ cm}^3 \implies \rho = \frac{54}{20} = 2.7\text{ g/cm}^3

Pressure, Force & Area

P = \frac{F}{A}, \quad F = P \times A, \quad A = \frac{F}{P}

P = pressure • F = force (Newtons) • A = cross-sectional area (m²).

Worked Example

\displaystyle F = 500\text{ N}, \; A = 2\text{ m}^2 \implies P = \frac{500}{2} = 250\text{ N/m}^2

Curriculum & Revision Note

About These Ratio & Proportion Formulas

Ratio and proportion questions feature prominently across secondary school exams. This sheet consolidates techniques for simplifying ratios, unitary forms 1 : n, sharing totals across parts, length-area-volume scale factors (k, k², k³), direct and inverse algebraic proportionality models (y = kx, y = k/x), alongside essential compound measure formulas (speed, density, pressure).

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