№08 · NUMBER · FOUNDATION + HIGHER
PRINT the A4 for your wall · SCREENSHOT the square for Instagram/Pinterest
A4 · 794 × auto · Print-ready
Σ
gcsemathsai
EST. MMXXIV · LONDON · MMXXVI SPEC.
№ 08 · VIII
NUMBER · FOUNDATION + HIGHER A4 · 210×297mm
Ratio & ProportionFoundation + Higher
Proportion. Direct & inverse.
Direct proportion: as one goes up, the other goes up at the same rate. Inverse proportion: one up, the other down. Both use a constant k — find k first, then use it.
I · Key definitions
Direct proportion
y goes up as x goes up, by the same factor.
y ∝ x → y = kx
Inverse proportion
y goes down as x goes up.
y ∝ 1/x → y = k/x
Constant k
The 'proportionality constant'. Find from given pair, then use.
if y=12 when x=3, k = 4
Proportional to square
y ∝ x² → y = kx²
area ∝ radius²
Unit method
Find value for one, then scale.
6 apples = £3 → 1 apple = £0.50 → 10 = £5
∝ symbol
Means 'is proportional to'.
y ∝ x reads "y is prop. to x"
II · The shape of the two graphs
Direct · y = kx
y = kx
straight line through origin
Inverse · y = k/x
y = k/x
reciprocal curve (hyperbola)
Direct = straight line passing through (0,0). Inverse = curve that never touches the axes.
III · The method — step-by-step
Direct proportion
y ∝ x → y = kx 1. Find k from given pair 2. Substitute k back 3. Use to answer question
y=15 when x=3 → k=5 → y = 5x
Inverse proportion
y ∝ 1/x → y = k/x 1. Find k from given pair 2. k = xy 3. Use formula
y=4 when x=3 → k=12 → y = 12/x
Proportional to square
y ∝ x² → y = kx²
y=20 when x=2 → k=5 → y = 5x²
Inverse square
y ∝ 1/x² → y = k/x²
y=9 when x=2 → k=36 → y = 36/x²
IV · Worked examples
Direct proportion
3 marks
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
i.Set up: y = kx
ii.Find k: 20 = k × 4 → k = 5
iii.Formula: y = 5x
iv.When x = 7: y = 5 × 7 = 35
y = 35 when x = 7
Inverse proportion
3 marks
4 people build a wall in 6 hours. How long would 8 people take, working at the same rate?
y is proportional to x². When x = 3, y = 45. Find y when x = 5.
i.y = kx²
ii.Find k: 45 = k × 9 → k = 5
iii.y = 5x² → when x = 5: y = 5 × 25 = 125
y = 125
V · Common mistakes & examiner tips
Common mistakes
Forgetting to find k first. Most important step — find k, then use it.
Direct vs inverse muddled. Read the question — 'faster' and 'fewer' usually = inverse.
Squaring after substituting k. k is a constant — only x changes.
Setting up wrong formula. 'Proportional to x² ' means y = kx², not y = (kx)².
Unit confusion. Ensure same units on both sides (mins vs hours, etc.).
Examiner tips
Write the formula first — y = kx or y = k/x — before doing anything.
Always show k explicitly. Substituting into y = 5x gets full method marks.
Sanity-check: if more workers → less time, that's inverse. If more time → more pay, that's direct.
For 'proportional to square': doubling x quadruples y.
Graphs: direct = straight line through origin; inverse = curve.
SQUARE · 900 × auto · Social-ready
Σ
gcsemathsai
EST. MMXXIV · LONDON · MMXXVI SPEC.
№ 08 · VIII
NUMBER · FOUNDATION + HIGHER Square · 1:1
Ratio & ProportionFoundation + Higher
Proportion. Direct & inverse.
Direct proportion: as one goes up, the other goes up at the same rate. Inverse proportion: one up, the other down. Both use a constant k — find k first, then use it.
I · Key definitions
Direct proportion
y goes up as x goes up, by the same factor.
y ∝ x → y = kx
Inverse proportion
y goes down as x goes up.
y ∝ 1/x → y = k/x
Constant k
The 'proportionality constant'. Find from given pair, then use.
if y=12 when x=3, k = 4
Proportional to square
y ∝ x² → y = kx²
area ∝ radius²
Unit method
Find value for one, then scale.
6 apples = £3 → 1 apple = £0.50 → 10 = £5
∝ symbol
Means 'is proportional to'.
y ∝ x reads "y is prop. to x"
II · The shape of the two graphs
Direct · y = kx
y = kx
straight line through origin
Inverse · y = k/x
y = k/x
reciprocal curve (hyperbola)
Direct = straight line passing through (0,0). Inverse = curve that never touches the axes.
III · The method — step-by-step
Direct proportion
y ∝ x → y = kx 1. Find k from given pair 2. Substitute k back 3. Use to answer question
y=15 when x=3 → k=5 → y = 5x
Inverse proportion
y ∝ 1/x → y = k/x 1. Find k from given pair 2. k = xy 3. Use formula
y=4 when x=3 → k=12 → y = 12/x
Proportional to square
y ∝ x² → y = kx²
y=20 when x=2 → k=5 → y = 5x²
Inverse square
y ∝ 1/x² → y = k/x²
y=9 when x=2 → k=36 → y = 36/x²
IV · Worked examples
Direct proportion
3 marks
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
i.Set up: y = kx
ii.Find k: 20 = k × 4 → k = 5
iii.Formula: y = 5x
iv.When x = 7: y = 5 × 7 = 35
y = 35 when x = 7
Inverse proportion
3 marks
4 people build a wall in 6 hours. How long would 8 people take, working at the same rate?