Key takeaways
- a/b = c/x → x = (b × c) ÷ a.
- Cross multiplication means multiplying diagonally across the equals sign.
- The same type of quantity must go on the same side of both fractions.
- The missing value can be a whole number, decimal, or fraction.
In this guide
What is a proportion?
A proportion is a statement that two ratios are equal, like "3 eggs to 5 cups of flour" is the same recipe ratio as "12 eggs to x cups of flour". Solving the proportion finds the missing value, x, that keeps both ratios equal.
Why does solving a proportion matter?
Proportions let you scale a known relationship to a new size, whether that's a recipe, a map distance, or a mixing ratio, without needing to redo the whole calculation from scratch.
Two ratios are proportional exactly when their cross products match: 2/3 and 8/12 are proportional because2 × 12 = 24and3 × 8 = 24.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a | The first term of the known ratio | 3 |
| b | The second term of the known ratio | 5 |
| c | The term you know in the second ratio | 12 |
| x | The missing value you're solving for | 20 |
How to solve a proportion step by step
- 1Set up the two ratios as equal fractions:
a/b = c/x, or3/5 = 12/x. - 2Cross multiply: multiply a by x, and multiply b by c:
3 × x = 5 × 12. - 3Set the two products equal:
3x = 60. - 4Divide both sides by a to isolate x:
x = 60 ÷ 3 = 20.
Enter your own known ratio and third term in the calculator below to see the full working.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Setting up the ratio in the wrong order. Make sure the same type of quantity (like eggs) is on top in both fractions, and the same type (like flour) is on the bottom in both.
- Forgetting to divide by a at the end. After cross multiplying, you still need to isolate x by dividing.
- Mixing up which value is missing. Double-check which term the problem is actually asking you to find.
Tips and tricks
- Simplify the known ratio first (see the ratio simplifier) if it has large numbers, to make the cross multiplication easier.
- Watch for scaling patterns. If c is a clean multiple of a, you can often just scale b by the same multiple without full cross multiplication.
- Once you understand cross multiplication here, you'll recognize it inside the percentage formula too, since percentages are just a special proportion out of 100.
Where you'll use it in real life
- Cooking: scaling a recipe up or down while keeping every ingredient in the same ratio.
- Maps: converting a map distance to a real-world distance using the map's scale.
- Shopping: working out the price of a different quantity of the same item.
- DIY and construction: scaling a mixing ratio, like concrete or paint, to a larger batch.
Quick summary
To solve a proportion, set the two ratios equal as fractions, cross multiply, then divide to isolate the missing value. Keep the same type of quantity on the same side of both fractions. Try the calculator above with your own proportion.
Worked example
A recipe uses 3 eggs for every 5 cups of flour. If a bakery wants to use 12 eggs and keep the same ratio, how many cups of flour do they need?
- 01a/b = c/x
- 023/5 = 12/x
- 03Cross multiply: 3 × x = 5 × 12
- 043x = 60
- 05x = 60 ÷ 3 = 20
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does cross multiplication mean?
It means multiplying diagonally across an equation of two fractions — the numerator of one side by the denominator of the other — to remove the fractions and solve for the unknown.
How do I know if two ratios form a true proportion?
Cross multiply both ratios. If the two products are equal, the ratios are proportional. For example, 2/3 and 8/12 are proportional because 2 × 12 = 24 and 3 × 8 = 24.
Can the missing value in a proportion be a decimal?
Yes. Proportions don't always give whole-number answers — the missing value can be a decimal or fraction, depending on the numbers involved.
What's the difference between a ratio and a proportion?
A ratio compares two quantities, like 3:5. A proportion is a statement that two ratios are equal, like 3:5 = 12:20, which you can use to solve for a missing value.
