Ratios, fractions and linear functions
A ratio a : b expresses a constant relationship between quantities. The same relationship can be written as a fraction, an equation, or a linear graph.
Ratio ↔ fraction
Ratio a : b means: for every a units of the first quantity, there are b units of the second.
- Fraction first / total = a / (a + b).
- Fraction first / second = a / b.
Example: ratio of boys : girls = 3 : 5. Fraction of boys = 3/8 of the class. Fraction "boys per girl" = 3/5.
Ratio ↔ linear function
If y : x = k : 1 (or equivalently y = kx), the relationship is a STRAIGHT LINE through the origin with gradient k.
- Direct proportion: y = kx ↔ ratio y : x is constant = k : 1.
This means: every WJEC ratio question can become a linear-function question, and vice versa.
Example
Cost C of x kg of apples is in the ratio C : x = 2 : 1 (so £2 per kg). Linear function C = 2x. The graph is a line through (0, 0) with gradient 2.
Sharing in a ratio
To share Q in the ratio a : b : c:
- Total parts = a + b + c.
- One part = Q ÷ total parts.
- Multiply by each share's parts.
Combining ratios
If A : B = 3 : 4 and B : C = 6 : 5, scale so B is the same in both.
- A : B = 9 : 12 (×3)
- B : C = 12 : 10 (×2)
- Therefore A : B : C = 9 : 12 : 10.
Comparing two ratios
Convert both to "per one unit" form (n : 1) by dividing both terms by the second.
WJEC exam tip
When a question says "in the ratio …", you almost always want to compute "one part" first. Writing "1 part = …" on its own line earns the M1 method mark even if subsequent arithmetic slips. Always check the answer makes sense — shares should add to the original total.
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