How to Solve Proportions (Direct and Inverse)
A proportion says that two ratios are equal. Whenever you know three values and the relationship between them, you can find the fourth. It is the math behind scaling recipes, comparing prices, converting rates and reading maps.
Direct proportions
Two quantities are directly proportional when they rise and fall together: twice as much of one means twice as much of the other.
Write it as A / B = C / X and solve with cross-multiplication:
X = B × C / A
Example: 3 pounds of apples cost $6. How much do 5 pounds cost? X = 6 × 5 / 3 = $10.
Scaling a recipe: a recipe for 4 servings uses 2 cups of flour. For 6 servings: X = 2 × 6 / 4 = 3 cups.
Map scale: 1 inch on a map equals 25 miles. Two towns are 3.2 inches apart: X = 25 × 3.2 / 1 = 80 miles.
Inverse proportions
Two quantities are inversely proportional when one goes up as the other goes down: twice as much of one means half as much of the other. The product stays constant, so A × B = C × X:
X = A × B / C
Example: 4 painters finish a house in 6 days. How long do 3 painters take? X = 4 × 6 / 3 = 8 days.
Speed and time: a trip takes 2 hours at 60 mph. At 75 mph: X = 60 × 2 / 75 = 1.6 hours, or 1 hour 36 minutes.
How to tell which one you have
Ask: if I double one quantity, what happens to the other?
| If you double… | …the other | Type |
|---|---|---|
| Pounds of apples | doubles (price) | Direct |
| Servings | doubles (ingredients) | Direct |
| Workers | halves (days needed) | Inverse |
| Speed | halves (travel time) | Inverse |
Unit prices: a practical use
Comparing a 24-ounce jar for $5.49 with a 32-ounce jar for $6.99? Scale both to the same size. The price of 32 ounces at the smaller jar’s rate is 5.49 × 32 / 24 = $7.32, so the larger jar is the better deal.
Percentages are proportions too
“What percent is 18 out of 24?” is the proportion 18 / 24 = X / 100, so X = 100 × 18 / 24 = 75%.
Common mistakes
- Using the direct formula for an inverse problem. If more of something should mean less of the result, switch formulas.
- Mixing units. Convert ounces to pounds or minutes to hours before solving.
- Assuming proportionality where there is none. Prices with a fixed fee, bulk discounts or travel with traffic do not scale exactly.
Results are estimates for informational purposes only and are not financial, tax, legal or medical advice.