Ratios as fractions and as linear functions

A ratio a : b can be re-expressed as a fraction (a/b or a/(a+b)) or as a linear function (y = (a/b)x). OCR J560 tests this connection on Higher papers especially, where ratios appear in graphs of direct proportion.

Ratio as fraction (recap)

a : b means "for every a units of one quantity, there are b of the other".

• "First as fraction of second" = a/b.
• "First as fraction of total" = a/(a+b).

Ratio as linear function

If two quantities x and y are in the ratio a : b, then y/x is constant: y/x = b/a.

This rearranges to y = (b/a) × x — a straight line through the origin with gradient b/a.

Example: "Height (cm) and weight (kg) of saplings are in ratio 2 : 1." So y = 0.5x — a saplings of x cm tall has weight 0.5x kg.

Using a graph

The graph of a direct-proportion ratio:

• Passes through (0, 0).
• Has gradient = ratio of (y-quantity to x-quantity).
• Any point (x, y) on the line satisfies the ratio.

Linking back to other forms

Form	Example for "y : x = 3 : 4"
Ratio	y : x = 3 : 4
Equation	y = (3/4)x
Fraction	y/x = 3/4
Words	"y is three-quarters of x"

Mixed examples

Q: "If apples and oranges in a fruit bowl are in ratio 5 : 3, and there are 32 fruits in total, how many of each?"

Method via fraction:

• Apples = 5/8 × 32 = 20.
• Oranges = 3/8 × 32 = 12.

Method via linear function:

• Let oranges = x. Then apples = (5/3)x.
• (5/3)x + x = 32 → (8/3)x = 32 → x = 12.
• Apples = 32 − 12 = 20.

Both methods give the same answer; the linear-function method generalises better when ratios appear in algebra problems.

OCR mark scheme conventions

• B1 for the correct ratio-to-fraction conversion.
• M1 for setting up a linear equation y = kx.
• A1 for the final values.
• For graph questions: B1 for line through origin with correct gradient.