Extended. When one quantity depends on another, through its square, its cube, its square root or its reciprocal, a single equation with one constant \(k\) captures the whole relationship.
Turn \(\propto\) into \(= k \times\). Direct: \(y = kx^n\). Inverse: \(y = \dfrac{k}{x^n}\). Use the given pair of values to find \(k\), write the formula, then use it.
Fig. 1: y = 1.2x (direct) is a straight line through the origin; y = 4/x (inverse) falls steeply then levels off, and never meets either axis
Section 01
Direct proportion.
In the lab, pick a type of proportion (\(y \propto x\), \(y \propto x^2\), ...) and set \(k\). Double \(x\) and read the new \(y\): for \(y \propto x\) it doubles, for \(y \propto x^2\) it goes up four times. Then try \(y \propto \dfrac{1}{x}\) and watch \(y\) halve.
In words
Symbol
Equation
\(y\) is proportional to \(x\)
\(y \propto x\)
\(y = kx\)
\(y\) is proportional to the square of \(x\)
\(y \propto x^2\)
\(y = kx^2\)
\(y\) is proportional to the cube of \(x\)
\(y \propto x^3\)
\(y = kx^3\)
\(y\) is proportional to the square root of \(x\)
\(y \propto \sqrt{x}\)
\(y = k\sqrt{x}\)
\(y\) is proportional to the cube root of \(x\)
\(y \propto \sqrt[3]{x}\)
\(y = k\sqrt[3]{x}\)
Worked example Linear
\(y\) is directly proportional to \(x\). When \(x = 4\), \(y = 12\). Find \(y\) when \(x = 7\).
Write the equation: \(y = kx\).
Substitute the known pair: \(12 = k \times 4\), so \(k = 3\).
Formula: \(y = 3x\). When \(x = 7\), \(y = 21\).
Answer: \(y = 21\)
Worked example Square
\(y \propto x^2\). When \(x = 5\), \(y = 50\). (a) Find \(y\) when \(x = 3\). (b) Find \(x\) when \(y = 128\), given \(x > 0\).
\(y = kx^2\), so \(50 = k \times 25\) and \(k = 2\). Formula: \(y = 2x^2\).
(a) \(y = 2 \times 3^2 = 18\).
(b) \(128 = 2x^2\), so \(x^2 = 64\) and \(x = 8\).
Answer: (a) 18 (b) 8
Examiner trap
Always find \(k\) and write the formula first. Scaling by eye ('x went from 5 to 3, so...') goes wrong as soon as there is a square or a root involved.
Section 02
Inverse proportion.
Inverse proportion means one quantity goes UP as the other goes DOWN, so that their product (or the product with a power) stays the same.
Worked example Reciprocal
\(y\) is inversely proportional to \(x\). When \(x = 4\), \(y = 6\). Find \(y\) when \(x = 3\).
\(y = \dfrac{k}{x}\), so \(6 = \dfrac{k}{4}\) and \(k = 24\).
Formula \(y = \dfrac{24}{x}\). When \(x = 3\), \(y = 8\).
Answer: \(y = 8\)
Worked example Square root
\(y\) is inversely proportional to the square root of \(x\). When \(x = 9\), \(y = 10\). Find \(y\) when \(x = 25\).
\(y = \dfrac{k}{\sqrt{x}}\), so \(10 = \dfrac{k}{3}\) and \(k = 30\).
When \(x = 25\): \(y = \dfrac{30}{5} = 6\).
Answer: \(y = 6\)
Your turn
\(y\) is inversely proportional to \(x^2\). When \(x = 2\), \(y = 5\). Find \(y\) when \(x = 4\).
\(y = \dfrac{k}{x^2}\): \(5 = \dfrac{k}{4}\), so \(k = 20\). When \(x = 4\), \(y = \dfrac{20}{16} = 1.25\).
Examiner trap
In \(y = \dfrac{k}{\sqrt{x}}\), substitute \(\sqrt{9} = 3\), not 9. Read the question for the exact power: 'the square of', 'the square root of' and 'the cube of' all change the equation.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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\(y \propto x\) and \(y = 15\) when \(x = 5\). Find \(y\) when \(x = 8\).
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\(y\) is inversely proportional to the square of \(x\). Which equation is correct?
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\(y \propto x^2\). If \(x\) is multiplied by 3, what is \(y\) multiplied by?
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Which graph shows that \(y\) is directly proportional to \(x\)?
Section 03
Proportion in context.
Science and real-life questions use proportion all the time. Name the variables as given, find \(k\) (it may carry units or be a decimal), and answer with units.
Worked example Falling object
The distance \(d\) metres that a stone falls is proportional to the square of the time \(t\) seconds. It falls 20 m in 2 s. (a) How far does it fall in 3.5 s? (b) How long does it take to fall 125 m?
\(d = kt^2\): \(20 = k \times 4\), so \(k = 5\) and \(d = 5t^2\).
The time a job takes is inversely proportional to the number of workers. 6 workers take 10 days. How long would 15 workers take?
\(T = \dfrac{k}{w}\): \(10 = \dfrac{k}{6}\), so \(k = 60\).
\(T = \dfrac{60}{15} = 4\) days.
Answer: 4 days
Your turn
\(y\) is proportional to the cube root of \(x\). When \(x = 8\), \(y = 6\). Find \(y\) when \(x = 125\).
\(y = k\sqrt[3]{x}\): \(6 = k \times 2\), so \(k = 3\). When \(x = 125\), \(y = 3 \times 5 = 15\).
Examiner trap
If \(y \propto x^2\) and \(x\) doubles, \(y\) is multiplied by \(2^2 = 4\), not 2. The multiplier for \(y\) is the multiplier for \(x\) raised to the same power.
Stage 2 · Exam
Exam-style questions
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\(y\) is proportional to the square root of \(x\). When \(x = 16\), \(y = 12\). Find \(y\) when \(x = 49\).
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\(y\) is inversely proportional to \(x^2\). When \(x = 4\), \(y = 3\). Find \(x\) when \(y = 12\), given \(x > 0\).
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The power \(P\) kW of a wind turbine is proportional to the cube of the wind speed \(v\) m/s. When \(v = 8\), \(P = 160\). Find \(P\) when \(v = 10\).
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\(y\) is inversely proportional to \(x\). \(x\) is increased by 25%. What happens to \(y\)?
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