Ratio
Simplifying Ratios
Divide all parts of the ratio by their HCF. Make sure all parts are in the same units first.
Worked Example
Simplify the ratio 45 : 60
HCF of 45 and 60 = 15
45 ÷ 15 : 60 ÷ 15 = 3 : 4
Sharing in a Given Ratio
1. Add the parts to find the total. 2. Divide the amount by the total parts. 3. Multiply by each part.
Worked Example
Share £200 in the ratio 3:2
Total parts = 3 + 2 = 5
One part = £200 ÷ 5 = £40
First share = 3 × £40 = £120
Second share = 2 × £40 = £80
Exam Tip
Always check your shares add up to the original total!
Direct & Inverse Proportion
Direct Proportion
y is directly proportional to x means y = kx. As x doubles, y doubles.
Can also be proportional to x², x³, or √x.
y ∝ x² means y = kx²
Find k by substituting known values.
Inverse Proportion
y is inversely proportional to x means y = kx. As x doubles, y halves.
Worked Example
y is inversely proportional to x. When x = 4, y = 6. Find y when x = 8.
y = k/x, so 6 = k/4, therefore k = 24
When x = 8: y = 24/8 = 3
Key Facts
- Direct proportion: graph is a straight line through the origin
- Inverse proportion: graph is a reciprocal curve (never touches the axes)
- The ∝ symbol means "is proportional to"
Growth & Decay
Compound Interest & Depreciation
Decay: Amount = P × (1 − r)n
where P = original amount, r = rate (as decimal), n = number of periods
Worked Example
£2000 is invested at 5% compound interest. Find its value after 3 years.
Amount = 2000 × (1.05)³
= 2000 × 1.157625
= £2315.25
Common Mistake
Don't confuse compound interest with simple interest. Compound interest earns interest on previous interest!
Compound Measures
Speed, Density & Pressure
Use the triangle method: cover what you want to find, and what's left is the formula.
Unit Conversions
km/h to m/s: × 1000 then ÷ 3600 (or ÷ 3.6)
m/s to km/h: × 3.6
Exam Tip
Always check your units match. If speed is in km/h, distance must be in km and time in hours.
Rates of Change
Interpreting Graphs
The gradient of a graph represents the rate of change.
- Distance-time graph: gradient = speed
- Velocity-time graph: gradient = acceleration, area under graph = distance
A steeper line means a faster rate of change. A horizontal line means no change (constant value).
Estimating Area Under a Curve
Split the area into trapeziums or rectangles and add them up. The more strips you use, the more accurate your estimate.
Key Facts
- A curved line on a distance-time graph means changing speed
- To find the speed at a point on a curve, draw a tangent and find its gradient
- A straight line on a velocity-time graph means constant acceleration
Ratio & Proportion Flashcards
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Ratio & Proportion Quiz
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Ratio - Mock Exam Questions
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