Method · Pre-Algebra
How to Solve a Proportion by Cross-Multiplying
A proportion says two ratios are equal. Cross-multiplying is not a trick — it is one multiplication applied to both sides.
What a proportion is
A proportion is a statement that two ratios are equal, written a/b = c/d. Knowing how to solve a proportion means finding whichever of those four values is missing, and it comes up far more often than the name suggests — scale drawings, recipe conversions, map distances, unit prices, similar triangles and most percentage questions are all one proportion in disguise.
The two fractions do not have to look alike. 3/4 and 9/12 are equal, so 3/4 = 9/12 is a true proportion even though none of the numbers match.
Cross-multiplying, and why it is allowed
To solve a/b = c/d, multiply both sides by bd. Every denominator cancels, and what is left is ad = bc — the products of the diagonal pairs.
- 3⁄4 = 9⁄xthe proportion
- 3x = 4 · 9cross-multiply: opposite products are equal
- 3x = 36evaluate the right side
- x = 12divide both sides by 3
Nothing unusual happened. Multiplying both sides of an equation by the same non-zero quantity is the ordinary move used on every equation; cross-multiplication is just the shape it takes when both sides are single fractions.
Setting the proportion up correctly
The arithmetic is easy; the setup is where answers go wrong. The rule is that both sides must be arranged the same way.
If a car travels 120 miles in 2 hours, and you want the distance in 5 hours, write miles over hours on both sides: 120/2 = x/5. Writing 120/2 = 5/x puts hours over miles on the right, and the answer comes out inverted.
- Label both sides with their units before writing any numbers.
- Check that the same quantity sits on top on both sides.
- Make sure the units match — 3 hours and 180 minutes only compare after conversion.
Solving for a value in the denominator
When the unknown is on the bottom, the method does not change. Cross-multiplying always produces a linear equation, whichever position the unknown occupies.
- 5⁄x = 8⁄24the proportion
- 5 · 24 = 8xcross-multiply
- 120 = 8xevaluate
- x = 15divide both sides by 8
Where proportions turn up
Recognising a proportion is most of the work in these problems. Some common forms:
| Situation | The proportion |
|---|---|
| Scale drawing at 1:50 | 1/50 = drawing length / real length |
| Unit price comparison | price / quantity = price / quantity |
| Percentage of a total | part / whole = percentage / 100 |
| Similar triangles | side / matching side = side / matching side |
The percentage form is worth remembering on its own. Almost any percentage question — finding a part, finding a whole, or finding the rate — is that single proportion with a different value missing.
Unit rates and scale factors
Two special cases of proportion appear so often they have their own names, and both are solved the same way.
- A unit rate is a ratio with 1 on the bottom — miles per hour, price per kilogram, words per minute. Finding one means solving a proportion where the denominator on the right is 1.
- A scale factor is the number every measurement is multiplied by. A 1:50 model has a scale factor of 50, and every real dimension is 50 times the drawing.
Unit rates are what make prices comparable. A 400 g jar at £2.40 and a 750 g jar at £4.20 are hard to compare directly, but the unit rates — £6.00 and £5.60 per kilogram — settle it immediately.
Equivalent ratios and equivalent fractions
Equivalent ratios are ratios that describe the same relationship with different numbers: 3:4, 6:8 and 30:40 are all equivalent. This is the same idea as equivalent fractions, because multiplying both parts of a ratio by the same number leaves the relationship untouched.
That is precisely why a proportion can be solved at all. Writing 3/4 = 9/x asserts that the two ratios are equivalent, and the solution is whichever value makes that true.
Direct proportion, and when it does not apply
Two quantities are in direct proportion when doubling one doubles the other. Distance at constant speed, total cost at a fixed unit price and ingredient amounts in a recipe all behave this way, and all can be solved with a single proportion.
Plenty of relationships do not. The area of a circle is not directly proportional to its radius — doubling the radius quadruples the area — so setting up a proportion there gives a confidently wrong answer. Check that the relationship really is proportional before reaching for the method.
Checking the answer
Substitute the value back and confirm the two cross-products match. In 3/4 = 9/12, the products are 3 × 12 = 36 and 4 × 9 = 36. If they differ, the setup was wrong, not the arithmetic.
A second, faster check: ask whether the answer is the right size. If the ratio on the left is less than 1, the ratio on the right must be too.
Questions about how to solve a proportion
Why does cross-multiplication work?
Because it is one legal move, not a shortcut. Multiplying both sides of a/b = c/d by bd clears both denominators and leaves ad = bc. The diagonal pattern is just what that multiplication looks like when written out.
Can I cross-multiply when there are three fractions?
Not directly. Cross-multiplication applies to one fraction equal to one fraction. Combine terms until each side is a single fraction first.
Is a ratio the same as a fraction?
A part-to-whole ratio is — 3 red out of 5 balls is 3/5. A part-to-part ratio, 3 red to 2 blue, is not; writing it as 3/2 describes something different.
What if cross-multiplying gives division by zero?
Then the proportion has no solution. A zero in the position you are dividing by means the two ratios cannot be equal for any value.