Solving proportions in one step
A proportion says that two ratios are equal: if 3 pounds of apples cost $7.50, then 5 pounds cost a matching amount. When three of the four numbers are known, the fourth follows by cross multiplication, a method often called the rule of three. This calculator solves both kinds of proportion: direct, where both quantities grow together, and inverse, where one shrinks as the other grows.
The formulas
Direct proportion: a/b = c/x, so x = b × c ÷ a. The ratio b ÷ a, the value of one unit, stays constant.
Inverse proportion: a × b = c × x, so x = a × b ÷ c. The product stays constant.
Worked example: direct
Three pounds of apples cost $7.50. How much do five pounds cost?
- Set up the proportion: 3 lb is to $7.50 as 5 lb is to x.
- Cross multiply: x = 7.50 × 5 ÷ 3 = $12.50.
- Check: the unit price is 7.50 ÷ 3 = $2.50 a pound, and 5 × 2.50 = 12.50.
| Pounds | Price |
|---|---|
| 1 | $2.50 |
| 2 | $5.00 |
| 3 | $7.50 |
| 5 | $12.50 |
| 10 | $25.00 |
| 12 | $30.00 |
Worked example: inverse
Four painters need 6 days to paint a house. How long would 3 painters take, working at the same pace?
- The total work is constant: 4 × 6 = 24 painter-days.
- With 3 painters: x = 4 × 6 ÷ 3 = 8 days.
Doubling the crew halves the time, which is the signature of an inverse proportion.
Where proportions show up
- Shopping: comparing a price per ounce, or working out what a different quantity will cost.
- Cooking: scaling a recipe from 4 to 6 servings means multiplying everything by 6 ÷ 4.
- Maps and models: on a 1:24,000 map, 1 inch stands for 24,000 inches, or 2,000 feet.
- Currency: if $1 buys €0.92, then $250 buys 250 × 0.92 = €230.
- Speed and time: at a fixed distance, driving faster shortens the trip in inverse proportion.
- Medicine and chemistry: dose per pound of body weight, or concentrations in a solution.
When the rule of three does not apply
The method works only when quantities are truly proportional. Some common traps:
- Fixed fees: a taxi with a $3 base fare plus $2 a mile is not proportional. Ten miles cost $23, not five times the cost of two miles.
- Volume discounts: a price that drops for larger quantities breaks the proportion.
- Growth that compounds: interest earned over several years grows faster than a straight line.
- Physical limits: 4 painters taking 6 days does not mean 24 painters finish in 1 day, because they would get in each other’s way.
Before using the result, ask whether doubling one quantity really doubles, or halves, the other.
Setting up the problem correctly
Write the two pairs in a small table, with matching units in each column: pounds above pounds, dollars above dollars. Then multiply diagonally and divide by the remaining number. Keeping units aligned is the simplest way to avoid putting a number in the wrong place.