Solving Proportions: Cross-Multiplication, Direct and Inverse
A proportion is a statement that two ratios are equal, such as 3/8 = x/20. If you know three of the four numbers, you can always find the fourth. That one skill answers a surprising range of questions: how much a larger batch will cost, how many items fit in a bigger space, or what a percentage means in real quantities.
This article shows the two standard ways to solve a proportion, explains the difference between direct and inverse proportion, and points out the situations where a proportion should not be used at all.
Cross-multiplication
If a/b = c/d, then multiplying both sides by b and d gives a × d = b × c. In practice, you multiply diagonally across the equals sign and set the two products equal.
a/b = c/d ⇒ a × d = b × cb and d must not be zero.
Worked example: solve 3/8 = x/20. Cross-multiply: 3 × 20 = 8 × x, so 60 = 8x and x = 60 ÷ 8 = 7.5. Check: 3/8 = 0.375 and 7.5/20 = 0.375.
Setting it up correctly matters more than the algebra. Keep the same quantity in the same position on both sides. If the left side is "notebooks over cost", the right side must be "notebooks over cost" too, not the other way round.
The unit-rate method
An alternative is to find the value of one unit, then scale it. It avoids setting up an equation and is easier to sanity-check.
Worked example: 5 notebooks cost 12.50. What do 8 cost? One notebook costs 12.50 ÷ 5 = 2.50, so eight cost 8 × 2.50 = 20.00. The cross-multiplication version, 5/12.50 = 8/x, gives 5x = 100 and x = 20. Both methods agree.
Percentages are proportions too. To find what percent 18 is of 45, write 18/45 = x/100. Cross-multiplying gives 45x = 1,800, so x = 40.
Direct versus inverse proportion
In direct proportion, two quantities rise and fall together at a constant ratio. Doubling the notebooks doubles the cost. The ratio of one to the other stays fixed.
In inverse proportion, one quantity rises as the other falls, and their product stays fixed. If 4 workers need 6 days to finish a job, the job holds 4 × 6 = 24 worker-days of effort. With 3 workers at the same pace, the time is 24 ÷ 3 = 8 days.
| Type | What stays constant | Example |
|---|---|---|
| Direct | Ratio y ÷ x | Cost of notebooks at a fixed price |
| Inverse | Product x × y | Workers and days, or speed and travel time over a fixed distance |
A second inverse example: a 120 km trip at 60 km/h takes 2 hours. At 80 km/h the time is 120 ÷ 80 = 1.5 hours. The product speed × time is always 120.
When a relationship is not proportional
Proportional reasoning only works if the relationship passes through zero: no input should mean no output. Many real situations include a fixed part, which breaks the assumption.
- A taxi that charges a 4.00 starting fee plus 2.00 per km is not proportional. A 5 km trip costs 14.00, but a 10 km trip costs 24.00, not 28.00.
- A phone plan with a monthly fee plus usage does not scale with the number of calls.
- A shop that gives a bulk discount past a certain quantity changes the price per item, so cost no longer scales in a straight line.
A quick test: check whether y ÷ x is the same for every pair of values. If the ratio changes, the relationship is not directly proportional, and a proportion will give a misleading answer. Another warning sign is when a fixed charge, such as a delivery fee, sits inside the total.
A checklist for setting up proportions
- Decide whether the relationship is direct or inverse.
- Label each quantity and keep the same order on both sides.
- Cross-multiply, or find the unit rate, and solve for the unknown.
- Check that the answer is sensible: more input should give more output in a direct proportion, and less in an inverse one.
- Substitute your answer back into the original ratio to confirm both sides match.
To check an answer or solve for any missing term, use the proportion calculator. The ratio calculator is useful when you only need to scale or simplify, and the percentage calculator covers proportion problems stated in percent.
- Proportion Calculator: Enter three of the four numbers in two equal fractions and leave one box empty. The calculator solves for the missing term and the guide explains the cross-multiplication rule behind it.
- Ratio Calculator: Fill in three of the four numbers in a:b = c:d, leave one empty and read off the missing value. The guide covers simplifying and combining ratios, odds and screen aspect ratios.
- Percentage Calculator: Pick the question you need answered, enter two numbers and get the result instantly. The guide below shows the formula behind each mode.
This article is general information, not professional advice. See our Disclaimer and Editorial Policy.
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