Rule of three calculator

Solve a proportion as you type

The rule of three finds a missing number when two quantities are in proportion: you know three values and want the fourth. This proportion calculator lays the numbers out in a small table, does the cross-multiplication and shows every step. The calculation runs in your browser: nothing is sent.

How to use the calculator

1. Choose "Direct proportion" (the usual case) or "Inverse proportion".
2. If you like, name the two quantities, for example "lb" and "$": the names are used in the result sentence.
3. Fill in three of the four boxes A, B, C and D and leave the one you are looking for empty. Any of the four can be the unknown. The answer appears in the empty box and under "Result".
4. Read the "Step-by-step working", then use "Copy the working", "Swap the rows" or "Clear all".

Next to "Examples", six buttons fill the table in one click, among them "Price per pound", "Map scale" and "Currency conversion" (type your own rate).

The method: cross-multiplication and the unitary method

Say 4 lb of apples cost $9. What do 7 lb cost? Write 4 and 9 on the first row, 7 and x on the second. In a proportion the cross products are equal: 4 × x = 9 × 7. So x = 9 × 7 ÷ 4 = 63 ÷ 4 = $15.75.

The unitary method reaches the same answer in two moves: 1 lb costs 9 ÷ 4 = $2.25, so 7 lb cost 2.25 × 7 = $15.75. Results are rounded to 4 decimal places. When your three numbers are whole and the division does not come out even, an "Exact value" line gives the fraction in lowest terms: with 3, 10 and 1 you read 3.3333 and 10/3.

Is it really proportional?

Two quantities are in direct proportion when you always multiply one by the same number to get the other. A quick test: if the first doubles, does the second double too? Fuel paid at the pump and distance covered at a steady speed pass. These do not, and the rule of three then gives a wrong answer:

• cooking time: a cake twice as big does not need twice as long in the oven;
• age and height: a 10-year-old who is 4 ft 7 in tall will not be 9 ft 2 in at 20;
• prices with a fixed part, such as a delivery fee or a taxi's starting fare.

Inverse proportion

Sometimes one quantity goes down when the other goes up, in the same ratio: their product stays constant. Example: 4 workers tile a floor in 6 days. With 8 workers, the job takes 4 × 6 ÷ 8 = 3 days. Through the unit: one worker alone would need 4 × 6 = 24 days, so 8 workers need 24 ÷ 8 = 3 days.

Everyday uses

• Recipes: the "Scale a recipe" block converts a whole list. Write one ingredient per line, quantity first, then fill in "Servings in the recipe" and "Servings wanted". From 4 to 6 servings, 200 g of flour becomes 300 g and 3 eggs become 4.5.
• Prices: bring two packs back to the same weight to compare them.
• Speed and time: 150 miles in 2.5 hours means 2.5 × 240 ÷ 150 = 4 hours for 240 miles.
• Percentages: 45 correct answers out of 60 is 45 × 100 ÷ 60 = 75 out of 100, or 75%.
• Scales: if 1 inch on the map stands for 2 miles, 3.5 inches stand for 7 miles.
• Household mixes: 30 ml of cleaner for 5 litres of water means 30 × 8 ÷ 5 = 48 ml for 8 litres.

Never work out a dose of medicine with an online tool: follow the prescription or ask a pharmacist.

Frequently asked questions

What is the rule of three formula?

If A goes with B and C goes with x, then x = B × C ÷ A. For an inverse proportion, x = A × B ÷ C.

Is this the same as a ratio calculator?

For the common question, yes. Solving the ratio 4 : 9 = 7 : x is the calculation above: put 4 and 9 in boxes A and B, 7 in box C and leave D empty.

What happens if I enter zero?

If the number you have to divide by is 0, there is no answer: the calculator shows a dash and names the box at fault. An entry that is not a number is flagged in the same way.

Why does my scaled recipe say 4.5 eggs?

The list shows the exact quantity. You decide: 4 large eggs or 5 small ones. A line without a quantity, such as "salt", is copied as it is.