Proportion & Cross Multiplication Calculator
Fill in three of the four values in A/B = C/D and get x with fully worked cross multiplication, exact fractions and decimals. Switch to inverse proportion, or check whether two ratios are equal.
Enter three values and leave the unknown box empty (or type x).
Enter two ratios to check whether they are equal (form a proportion).
Step-by-step working
Word problem examples
How to use the Proportion & Cross Multiplication Calculator
- Choose Direct or Inverse. Direct means both quantities grow together; inverse means one goes up as the other goes down.
- Enter three values and leave the unknown box empty, or type x in it.
- Read x as a fraction, mixed number and decimal, with every cross multiplication step shown.
- To test two ratios, open Check a proportion and enter all four numbers.
- Try a word problem from the examples to see how to set one up.
The cross multiplication formulas
Starting from A / B = C / D, cross multiplication gives A × D = B × C. Rearranging gives a formula for each position of the unknown:
| Unknown | Formula | Example |
|---|---|---|
| A | A = B × C ÷ D | x/8 = 9/12 → x = 8 × 9 ÷ 12 = 6 |
| B | B = A × D ÷ C | 5/x = 15/24 → x = 5 × 24 ÷ 15 = 8 |
| C | C = A × D ÷ B | 3/4 = x/20 → x = 3 × 20 ÷ 4 = 15 |
| D | D = B × C ÷ A | 2/7 = 10/x → x = 7 × 10 ÷ 2 = 35 |
For an inverse proportion, x₁ × y₁ = x₂ × y₂, so the unknown is the known product divided by its partner. If 6 machines finish a job in 10 hours, 4 machines need 6 × 10 ÷ 4 = 15 hours.
Direct or inverse? How to decide
- Direct proportion: more of one means more of the other at the same rate. Examples: price and quantity, distance and fuel used, ingredients and servings, map distance and real distance, hours worked and pay.
- Inverse proportion: more of one means less of the other, and their product stays fixed. Examples: workers and time to finish a job, speed and travel time for a fixed distance, number of people sharing a cost and cost per person.
- Neither: many real relationships are not proportional at all, such as a taxi fare with a fixed starting charge or age and height. Proportions only work when the ratio or the product really is constant.
Setting up a word problem
Keep the same units in the same positions on both sides. A good habit is to write the units in words first: cups / cookies = cups / cookies. Then fill in the numbers you know and put x where the unknown goes.
Example: "A car uses 6 liters of fuel to drive 80 km. How much fuel for 200 km?" Set up 6 / 80 = x / 200. Cross-multiply: 80x = 6 × 200 = 1,200, so x = 1,200 ÷ 80 = 15 liters. A quick sanity check: 200 km is 2.5 times 80 km, and 2.5 × 6 = 15.
Percentages are proportions too: "What is 35% of 240?" is 35 / 100 = x / 240, which gives x = 84.
Frequently asked questions
What is a proportion?
A proportion is an equation stating that two ratios are equal, such as 3/4 = 15/20. Proportions let you find a missing value when two quantities change at the same rate.
How does cross multiplication work?
In A/B = C/D, multiplying both sides by B and D gives A × D = B × C. If one value is unknown, you divide the product of the two known diagonal values by the remaining known value. For 3/4 = x/20, 4x = 60, so x = 15.
What is the difference between direct and inverse proportion?
In a direct proportion the ratio stays constant: double one quantity and the other doubles (cost and number of items). In an inverse proportion the product stays constant: double one and the other halves (workers and days to finish a job).
How can I tell if two ratios form a proportion?
Cross-multiply. If A × D equals B × C, the ratios are equal and form a proportion. For 2/3 and 10/15, 2 × 15 = 30 and 3 × 10 = 30, so they are proportional.
Why does the answer show a fraction and a decimal?
Many proportions do not give a whole number. The simplified fraction is exact, while the decimal may repeat forever. A bar over digits, as in 0.3 with a bar over the 3, means those digits repeat.
Can I use decimals or fractions as inputs?
Yes. Enter values like 2.5, 3/4 or 1 1/2. The calculator works with exact fractions internally, so there is no rounding error in the steps.