Direct Proportion and Cross-Multiplication

Q: Why does cross-multiplication work?

It is a shortcut for multiplying both sides of a/b = c/d by b and by d, which clears both denominators and leaves ad = bc.

How to set up and solve a proportion by cross-multiplying, why it works, and how direct proportion shows up as a straight-line graph through the origin.

3 min read Supports 3 lessons

If $3$ notebooks cost £7.50, how much do $5$ cost? You probably reach for a unit price. A proportion is another way: it states that the ratio of notebooks to cost is the same for both purchases.

\frac{3}{7.50} = \frac{5}{x}

Cross-multiplication

\frac{a}{b} = \frac{c}{d} \;\Longleftrightarrow\; ad = bc

Why is this allowed? Multiply both sides of the equation by $b$ and by $d$ . The $b$ cancels on the left and the $d$ cancels on the right. Cross-multiplying is just a shortcut for those two steps.

$\frac{4}{x} = \frac{10}{25}$
Solve for $x$ .

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Worked example

If $3$ notebooks cost £7.50, how much do $5$ cost?

1
Set up the proportion: notebooks over cost.
$\frac{3}{7.50} = \frac{5}{x}$

Watch out

Keep the same order on both sides. If the left is notebooks over cost, the right must be notebooks over cost too.

Direct proportion

Two quantities are directly proportional when doubling one doubles the other, tripling one triples the other, and so on. Their ratio never changes, so one is always a fixed multiple of the other.

Direct proportion

y = kx

$k$ is the constant of proportionality: the unit rate.

Directly proportional quantities lie on a straight line through the origin.

Key idea

A proportional relationship has a straight-line graph that passes through $(0, 0)$ . A line that misses the origin, like a taxi fare with a fixed starting charge, is linear but not proportional.

Tip

To test for direct proportion, divide $y$ by $x$ for each pair. If you get the same answer every time, the quantities are proportional and that answer is $k$ .

Inverse proportion, briefly

Some quantities move in opposite directions. If $4$ painters take $6$ days, $8$ painters take $3$ days: double one and the other halves. Here the product stays constant, $xy = k$ . That is called inverse proportion.

Common questions

Why does cross-multiplication work?: It is a shortcut for multiplying both sides of $\frac{a}{b} = \frac{c}{d}$ by $b$ and by $d$ , which clears both denominators and leaves $ad = bc$ .
How can I tell if a graph shows direct proportion?: It must be a straight line that passes through the origin. A straight line with a non-zero intercept is linear but not proportional.

Practise proportion

The Proportions lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Why does cross-multiplication work?

It is a shortcut for multiplying both sides of $\frac{a}{b} = \frac{c}{d}$ by $b$ and by $d$ , which clears both denominators and leaves $ad = bc$ .

How can I tell if a graph shows direct proportion?

It must be a straight line that passes through the origin. A straight line with a non-zero intercept is linear but not proportional.