Proportion Equations GCSE: Direct and Inverse Proportion Using k

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:

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?

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.

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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