Multiplicative Relationships

A multiplicative relationship exists when one quantity is always a fixed multiple of another. Recognising this allows the relationship to be expressed as a ratio, a fraction, or a linear equation of the form \( y = kx \). This connects ratio, proportion, and the gradient of a straight-line graph through the origin.

Three Equivalent Forms

If quantity B is always 3 times quantity A, this single multiplicative relationship can be written in three ways:

As a ratio: \( A : B = 1 : 3 \)
As a fraction: \( \frac{A}{B} = \frac{1}{3} \) or \( \frac{B}{A} = 3 \)
As an equation: \( B = 3A \) (a linear function with gradient 3 through the origin)

These three forms are interchangeable — moving fluently between them is the key skill.

Finding the Multiplier

Given two related quantities, the multiplier \( k \) is found by dividing one by the other:

\[ k = \frac{B}{A} \qquad \text{so that } B = kA \]

This multiplier is also the scale factor of an enlargement, the constant of proportionality, and the gradient of the line \( y = kx \).

Comparison Using Multiplicative Reasoning

To compare two quantities multiplicatively, ask: "how many times larger is one than the other?" rather than "what is the difference?" Multiplicative comparisons are more informative when the baseline quantities differ.

Example: shop A sells 3 items for £5; shop B sells 5 items for £7. Compare value: unit prices are \( \frac{500}{3} \approx 167 \)p and \( \frac{700}{5} = 140 \)p. Shop B is cheaper per item — the multiplicative comparison \( \frac{167}{140} \approx 1.19 \) shows shop A charges about 19% more per item.

Worked Examples

A car travels 240 miles on 8 gallons. Express this as (a) a ratio, (b) a fraction, (c) an equation.

(a) Miles : gallons \( = 240 : 8 = 30 : 1 \)

(b) \( \frac{\text{miles}}{\text{gallons}} = 30 \) (miles per gallon)

In a recipe, sugar and flour are in the ratio 2 : 5. Express sugar as a fraction of flour. If flour = 350 g, find sugar.

\[ \frac{\text{sugar}}{\text{flour}} = \frac{2}{5} \qquad \text{sugar} = \frac{2}{5} \times 350 = 140 \text{ g} \]

A map uses scale 1 : 25 000. Express the multiplier as a fraction and as a decimal.

\[ \text{Map distance} = \frac{1}{25{,}000} \times \text{real distance} = 0.00004 \times \text{real distance} \]

Tin A: 400 g for £1.20. Tin B: 600 g for £1.68. Which is better value? Express as a multiplicative comparison.

Unit price A: \( \frac{120}{400} = 0.3 \)p/g. Unit price B: \( \frac{168}{600} = 0.28 \)p/g.

\[ \frac{0.3}{0.28} \approx 1.07 \]

Tin A costs about 7% more per gram — Tin B is better value.

Key Takeaways

A multiplicative relationship \( B = kA \) can be expressed as ratio \( A : B = 1 : k \), fraction \( \frac{B}{A} = k \), or equation \( y = kx \).
The multiplier \( k \) is the constant of proportionality, the scale factor, and the gradient of \( y = kx \).
To compare value: find unit price (or unit rate) for each option; a multiplicative comparison shows how many times more expensive one is than the other.
The three representations (ratio, fraction, equation) are equivalent — translating between them is a core algebraic fluency skill.