Ratios, fractions and linear functions
A ratio is a tidy way to capture a constant multiplicative relationship — exactly the kind of relationship described by a linear function y = kx through the origin. This topic links three views of the same idea.
Ratio ↔ fraction ↔ y = kx
If A : B = a : b, then A/B = a/b. If we let A = y and B = x, the relationship becomes: y = (a/b) × x, a straight-line graph through the origin with gradient a/b.
So every constant-rate problem (recipes, exchange rates, density, speed) is really a linear function in disguise.
Worked example: 1 kg costs £3.50. The ratio cost : weight = 3.5 : 1, or written as a function: C = 3.5W. The graph has gradient 3.5.
Why "through the origin"
Direct proportion means doubling x doubles y. So when x = 0, y = 0 — there's no fixed offset. This is what makes it a "ratio relationship" rather than a more general linear function with intercept c ≠ 0.
From ratio table to graph
| Number of apples (n) | Cost (£) |
|---|---|
| 1 | 0.40 |
| 2 | 0.80 |
| 5 | 2.00 |
Function: C = 0.40 n. Graph: straight line, gradient 0.4, through (0, 0).
Inverse vs direct
If A : B = a : b is fixed, that's direct proportion: A = (a/b) B. If AB = constant (so as A doubles, B halves), that's inverse proportion: A = k/B.
Scaling within a ratio
If recipe A : B = 3 : 5 and you double the recipe, you get 6 : 10 — same ratio, just scaled. The function A = (3/5)B is unchanged.
⚠Common mistakes
- Adding to both sides — 3 : 5 ≠ 4 : 6. Ratios must be multiplied/divided, not added.
- Confusing y = kx (proportion) with y = mx + c (general linear) — only y = kx is a ratio relationship.
- Misreading "y is k times x" as additive — it's multiplicative.
- Forgetting that ratios can become fractions of total — 3 : 5 doesn't mean 3/5; it means 3/8 of total.
- Reading the graph gradient backwards — gradient = y / x = a / b in ratio form.
➜Try this— Quick check
A ratio of grams to litres is 250 : 1. Express as a function. Answer: g = 250 ℓ (linear, through origin).
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