Sheet № 116 · Higher only · AQA · Edexcel · OCR
Direct Proportion Equations –
Direct proportion equations are a Higher-tier topic that links algebra with ratio and proportion. When two quantities are directly proportional, one is always a constant multiple of the other. Understanding how to set up, find, and use the equation connecting them is essential for securing marks on this topic.
§Key definitions
Question:
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
Answer:
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Q1 (Foundation):
y is directly proportional to x. When x = 6, y = 15. Find y when x = 10.
Q2 (Foundation):
y ∝ x. When x = 8, y = 24. Find x when y = 42.
Q3 (Higher):
y is directly proportional to the cube of x. When x = 2, y = 48. Find y when x = 3.
§Formulas to memorise
y ∝ x gives y = kx
y ∝ x² gives y = kx²
y ∝ √x gives y = k√x
Write the proportionality statement — using ∝ (e.g., y ∝ x²).
Replace ∝ with = k — to form the equation (e.g., y = kx²).
Substitute — the given pair of values to find k.
Rewrite — the equation with the value of k.
Use — the equation to answer the question.
Worked example
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
Working: y ∝ x, so y = kx. Substitute x = 4, y = 20: 20 = k × 4, so k = 5. Equation: y = 5x. When x = 7: y = 5 × 7 = 35.
⚠ Common mistakes
- ✗Forgetting to square (or square-root) x before substituting. If y ∝ x², the equation is y = kx², not y = kx. Read the proportionality statement carefully.
- ✗Using the wrong type of proportion. Direct proportion means y = kx^n; inverse proportion means y = k/x^n. Check whether y increases or decreases as x increases.
- ✗Not finding k first. Some students try to jump directly to the answer. Always find k explicitly — it earns method marks and prevents errors.
✦ Exam tips
- →Always write the proportionality statement and the equation with k as separate lines. Examiners award marks for each step.
- →If the question says "y is proportional to the square root of x", write y ∝ √x, then y = k√x.
- →Check your answer makes sense: if y is proportional to x² and x increases, y should increase by a larger factor.