Direct proportion (cross multiplication)
Both quantities grow together: double one and the other doubles. Price by weight, recipe by serving.
- Proportion2 : 10 = 5 : X
- Cross multiplication2 × X = 10 × 5
- Product B × C50
- X = (B × C) ÷ A50 ÷ 2
- X25
How proportions work
Two quantities are proportional when a change in one causes a predictable change in the other. To solve a proportion, write the ratio from the known values, cross multiply, and isolate the unknown, called X.
The Direct tab covers price by weight, scaling a recipe, gas use, and average speed. The Inverse tab covers cases where one side rises and the other falls, like workers and days. The Compound tab combines two quantities in one problem. For percent problems, the percentage calculator applies the same idea with a base of 100.
Direct or inverse: how to tell
Ask yourself: if one quantity doubles, what happens to the other? If it doubles too, the proportion is direct. If it drops by half, the proportion is inverse.
| Situation | Relationship | Why |
|---|---|---|
| Pounds of apples and price paid | Direct | More pounds, more dollars |
| Painters and days to finish the house | Inverse | More people, fewer days |
| Driving speed and trip time | Inverse | Faster means less time |
| Gallons of gas and miles driven | Direct | More gallons, more miles |
| Faucets open and time to fill a tub | Inverse | More faucets, less time |
Worked examples
| Problem | Type | Math | Answer |
|---|---|---|---|
| 2 lb cost $10. How much do 5 lb cost? | Direct | 10 × 5 ÷ 2 | $25 |
| 4 painters take 6 days. How long for 3 painters? | Inverse | 4 × 6 ÷ 3 | 8 days |
| 3 machines at 4 h/day make 120 parts. What about 6 machines at 2 h/day? | Compound (both direct) | 120 × (6÷3) × (2÷4) | 120 parts |
| 4 workers at 6 h/day finish a deck in 8 days. What about 8 workers at 4 h/day? | Compound (both inverse) | 8 × (4÷8) × (6÷4) | 6 days |
| 1 gallon of paint covers 350 ft². How many ft² do 3 gallons cover? | Direct | 350 × 3 ÷ 1 | 1,050 ft² |
In the third example, doubling the machines doubles output, but cutting the hours in half brings output back to the starting point. For areas and room measurements, see the square footage calculator.
Common mistakes
- Using a direct proportion on an inverse problem. Workers and days is the classic trap.
- Mixing units. Convert minutes to hours first so both sides of the ratio use the same unit.
- Swapping quantities in the setup. Keep the same quantity in the same column on both sides.
- Forcing a proportion onto a situation that has none, like age and height.
For ratios written as fractions, use the fraction calculator.
Frequently asked questions
How do you solve a proportion with cross multiplication?
Write A is to B as C is to X, then cross multiply: X = (B × C) ÷ A. If 2 lb cost $10, then 5 lb cost 10 × 5 ÷ 2 = $25.
What is the difference between direct and inverse proportion?
In a direct proportion, both quantities grow together and X = (B × C) ÷ A. In an inverse proportion, one grows while the other shrinks and X = (A × B) ÷ C. Workers and days make the classic inverse case.
How do you solve a compound proportion?
Start from the known result and multiply by one factor per quantity: new ÷ old for direct, old ÷ new for inverse. This calculator handles two quantities.
How do I know if a proportion is direct or inverse?
Double one quantity in your head. If the other doubles too, it is direct. If the other is cut in half, it is inverse. Always compare each quantity with the result.
Can I use a proportion to find a percentage?
Yes. A percent is a direct proportion with a base of 100: if 200 equals 100%, then 50 equals X% with X = 50 × 100 ÷ 200 = 25%. The percentage calculator automates these cases.
Sources and method
- Ratios and proportional relationships (middle school math, Common Core State Standards)