How to solve proportions without mistakes
Utilio editorial teamPublished: 6 min read
When two quantities are proportional, you can find an unknown value from three known ones. If 3 lb of apples cost $7.50, then 5 lb cost 7.50 × 5 ÷ 3 = $12.50. This method is often called the rule of three, or cross-multiplication. The arithmetic is easy; the hard part is setting it up correctly: knowing whether the proportion is direct or inverse, and checking that the quantities really are proportional.
Set it up in two rows
Write what you know in two rows, one per situation, with the same kind of quantity in the same column. The unknown (X) goes in the second row:
| Pounds | Price | |
|---|---|---|
| Known situation | 3 | $7.50 |
| New situation | 5 | X |
Before calculating, ask one question: if the first quantity goes up, does the second go up or down? More pounds cost more money, so this is a direct proportion.
Direct proportion
When both quantities grow together (if one doubles, so does the other), multiply across and divide by the remaining number:
Direct: A corresponds to B; C corresponds to X
X = B × C ÷ A
Example: scaling a recipe
A recipe for 4 servings uses 200 g of flour. How much flour do you need for 6 servings?
More servings, more flour: it’s direct.
X = 200 × 6 ÷ 4 = 300 g
Inverse proportion
When one quantity goes up and the other goes down by the same factor (if one doubles, the other halves), multiply the two values in the known row and divide by the third:
Inverse: A corresponds to B; C corresponds to X
X = A × B ÷ C
Example: hoses filling a pool
Two identical hoses fill a pool in 6 hours. How long do three of those hoses take?
More hoses, less time: it’s inverse.
X = 2 × 6 ÷ 3 = 4 hours
Had you used a direct proportion here, you’d get 9 hours: more hoses would take longer, which makes no sense. Asking whether the answer is reasonable is the best way to catch the wrong setup.
Enter the three values, choose direct or inverse, and get the answer with the formula used.
Open the ratio calculatorPercentages as proportions
A percentage is simply a direct proportion where one of the quantities is 100, so it solves the three classic questions:
| Question | Setup | Answer |
|---|---|---|
| What is 15% of 80? | 100 → 80; 15 → X | X = 80 × 15 ÷ 100 = 12 |
| What percent is 18 of 24? | 24 → 100; 18 → X | X = 100 × 18 ÷ 24 = 75% |
| $45 is 30% of the total. What’s the total? | 30 → 45; 100 → X | X = 45 × 100 ÷ 30 = $150 |
For these you can also use the Percentage calculator, which has all three cases ready.
Problems with more than two quantities
When three or more quantities are involved (sometimes called a compound rule of three), solve it in steps: change one quantity at a time and decide at each step whether it’s direct or inverse.
Example: workers, hours and days
5 workers, working 8 hours a day, finish a job in 12 days. How many days will 6 workers take working 10 hours a day?
Step 1, change the workers (5 to 6). More workers, fewer days: inverse. 12 × 5 ÷ 6 = 10 days.
Step 2, change the hours per day (8 to 10). More hours per day, fewer days: inverse. 10 × 8 ÷ 10 = 8 days.
It takes 8 days
When proportions don’t work
This method only works when quantities really are proportional. Plenty of everyday situations aren’t, and cross-multiplying gives an answer that looks precise but is wrong:
Signs that the quantities aren’t proportional
- There’s a fixed part. A taxi or rideshare with a base fare plus a per-mile rate: 10 miles don’t cost twice as much as 5 miles.
- There are tiers or volume discounts. With tiered utility rates or bulk pricing, using twice as much doesn’t cost exactly twice as much.
- There are physical limits. If a cake bakes in 40 minutes, two cakes in the same oven don’t take 80 minutes.
- The work can’t be split. Adding people doesn’t always finish faster: some tasks can’t be divided.
Frequently asked questions
Why is it called the rule of three?
Because three known values are used to find a fourth, unknown one.
How do I know whether it’s direct or inverse?
Imagine doubling the first quantity. If the second also doubles, it’s direct. If it halves, it’s inverse. If it does neither exactly, a proportion isn’t the right tool.
Does it work with decimals?
Yes. For example, if 2 gallons of paint cover 700 sq ft, then 1,050 sq ft need 2 × 1,050 ÷ 700 = 3 gallons.
Tools in this guide
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