Proportion equations use algebra to describe how two quantities vary together. For direct proportion, y = kx; for inverse proportion, y = k/x. At GCSE you go beyond spotting proportional patterns and instead form an equation, calculate k, and use it to find unknown values precisely.
What is the proportionality symbol and how do you convert it to an equation?
The symbol ∝ means "is proportional to." The proportionality statement is the starting point for every proportion equation.
To convert a proportionality statement into an equation, replace ∝ with = k, where k is the constant of proportionality:
$$y \propto x \longrightarrow y = kx$$
Once you know k, the equation is fully defined and you can calculate any value of y from any value of x.
What types of direct proportion appear at GCSE?
Direct proportion can involve any power of x. The most common are:
| Proportionality | Equation | Graph shape |
|---|---|---|
| y ∝ x | y = kx | Straight line through origin |
| y ∝ x² | y = kx² | Parabola through origin |
| y ∝ x³ | y = kx³ | Cubic through origin |
| y ∝ √x | y = k√x | Square-root curve through origin |
For all of these, when x doubles, y changes by the same predictable factor: doubling x doubles y (for y ∝ x), quadruples it (y ∝ x²), octuples it (y ∝ x³), and multiplies it by √2 (y ∝ √x).
How do you find k from a pair of values?
Worked example 1 — Direct proportion: y is directly proportional to x². When x = 4, y = 48. Find the equation connecting y and x.
Step 1: Write the equation form. $$y = kx^2$$
Step 2: Substitute the known pair. $$48 = k \times 4^2 = 16k$$ $$k = 48 \div 16 = 3$$
Step 3: Write the complete equation. $$y = 3x^2$$
Step 4: (If asked) find y when x = 5: y = 3 × 25 = 75.
What types of inverse proportion appear at GCSE?
| Proportionality | Equation |
|---|---|
| y ∝ 1/x | y = k/x |
| y ∝ 1/x² | y = k/x² |
| y ∝ 1/√x | y = k/√x |
Worked example 2 — Inverse proportion: y is inversely proportional to √x. When x = 9, y = 4. Find y when x = 25.
Step 1: Write the equation. $$y = \frac{k}{\sqrt{x}}$$
Step 2: Substitute the known pair (x = 9, y = 4). $$4 = \frac{k}{\sqrt{9}} = \frac{k}{3}$$ $$k = 12$$
Step 3: Complete equation: y = 12/√x.
Step 4: Find y when x = 25. $$y = \frac{12}{\sqrt{25}} = \frac{12}{5} = \mathbf{2.4}$$
How do you identify the type of proportion from a table?
Given a table of values, check which relationship keeps a ratio constant:
- If y/x = constant → y ∝ x (direct, linear)
- If y/x² = constant → y ∝ x²
- If y × x = constant → y ∝ 1/x
- If y × x² = constant → y ∝ 1/x²
Example: Determine the type of proportion from this table.
| x | 1 | 2 | 4 | 5 |
|---|---|---|---|---|
| y | 100 | 25 | 6.25 | 4 |
Check y × x: 100×1 = 100, 25×2 = 50 (not constant). Check y × x²: 100×1 = 100, 25×4 = 100, 6.25×16 = 100, 4×25 = 100. All equal 100.
So y ∝ 1/x², and k = 100, giving the equation y = 100/x².
How do you find both variables when given extra information?
Some GCSE problems give you enough information to find k and then ask for x (not just y).
Worked example: y ∝ x³. When x = 2, y = 24. Find x when y = 192.
Step 1: Find k: 24 = k × 8, so k = 3. Equation: y = 3x³.
Step 2: Substitute y = 192: $$192 = 3x^3$$ $$x^3 = 64$$ $$x = \sqrt[3]{64} = \mathbf{4}$$
When solving for x, the inverse operation depends on the power: use square root for x², cube root for x³, and squaring for √x.
How do proportion equations appear in context?
GCSE questions often set proportion in a physical context:
- Kinetic energy is proportional to v² (speed squared).
- Gravitational force between two masses is inversely proportional to d² (distance squared) — Newton's law of universal gravitation.
- Period of a pendulum is proportional to √l (length).
Example: The time T seconds for a pendulum to complete one swing is proportional to √l, where l is the length in centimetres. When l = 25, T = 1. Find T when l = 100.
T = k√l. From l = 25, T = 1: 1 = k√25 = 5k, so k = 0.2.
T = 0.2√100 = 0.2 × 10 = 2 seconds.
Frequently asked questions
What is the difference between y = kx and y ∝ x?
y ∝ x is a statement that the relationship is proportional — it does not specify the value of k. y = kx is the equation form, which is only complete once you know k. In an exam, you must always find k from given data before using the equation to find unknowns.
Can the constant k be a fraction or decimal?
Yes — k can be any non-zero value, including fractions and decimals. It is defined entirely by the data. If the numbers in the question look "awkward" and you end up with k = 1/3 or k = 0.08, that is perfectly fine — substitute back to check your answer makes sense.
How do I avoid confusing the different types of proportion?
Write the proportionality statement (y ∝ …) before converting to an equation. The statement tells you the function of x on the right-hand side. Once you have written y = k × (function of x), the rest is substitution and algebra. The most common confusion is between y ∝ x² (parabola) and y ∝ 1/x² (inverse square) — check whether the question says "proportional to x squared" or "inversely proportional to x squared."
Does this topic overlap with compound interest or speed–distance–time?
Proportion equations are a separate formal topic from compound interest, though both involve multiplicative relationships. Speed–distance–time uses the formula v = d/t, which looks like an inverse proportion (d ∝ 1/t at constant v) but is usually solved by substitution rather than the k-method. The k-method is specifically for exam questions that use the ∝ notation and ask you to form an equation.
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