What is a proportion?
A proportion is a statement that two ratios are equal, written as a/b = c/d. If 3 out of 4 pieces are red in one pattern, a proportion helps you find how many red pieces you would expect in a bigger pattern with the same mix.
Proportions are used for scaling recipes, reading maps, converting units, working out prices per item, and many geometry problems with similar shapes.
How to solve a proportion by cross multiplication
- Write the proportion as a/b = c/d.
- Cross multiply: a ร d = b ร c.
- Isolate the unknown by dividing both sides by the number next to it.
Using the calculator
- Fill in three boxes and leave one empty. The calculator knows which one to solve.
- Decimals and negative numbers work.
- If you fill in all four boxes, or leave two or more empty, you will see a message asking you to leave exactly one empty.
- The denominators b and d cannot be zero.
Real-world examples
| Situation | Proportion |
|---|---|
| Recipe: 2 cups serve 6, how many for 15? | 2/6 = x/15, so x = 5 |
| Map scale: 2 in = 50 mi, how far is 7 in? | 2/50 = 7/x, so x = 175 |
| Price: 4 items cost $10, cost of 10? | 4/10 = 10/x, so x = 25 |
In each case the two ratios must compare the same kinds of things in the same order.
Simplified ratios
A ratio like 12 : 16 can be written more simply as 3 : 4 by dividing both numbers by their greatest common divisor, which is 4 here. The calculator does this for the a : b and c : d pairs. If your values are decimals, it first scales them to whole numbers when it can, then simplifies.
Common mistakes
- Putting the numbers in a different order on each side. If the left side is red over total, the right side must also be red over total.
- Cross multiplying the wrong pairs. Multiply across the diagonal: a with d, b with c.
- Mixing units, such as minutes on one side and hours on the other.
- Dividing by zero when a denominator is 0.
If you need to compare two numbers by percent instead, see the percentage difference calculator.
Direct and inverse proportion
The calculator solves a direct proportion, where both quantities grow or shrink together, such as more items costing more money. An inverse proportion works the other way: if more workers finish a job, it takes fewer days. For an inverse relationship the products stay equal (workers ร days is constant), so set up a/b = d/c instead, or swap one side before you enter the numbers.
Proportions with decimals and fractions
Fractions can be entered as decimals. To solve 1.5/4 = x/10, cross multiply to get 1.5 ร 10 = 4 ร x, so x = 15 รท 4 = 3.75. The calculator accepts any real numbers, so you do not need to clear decimals first.
Proportions in similar shapes and scale drawings
When two shapes are similar, matching sides are in proportion. If a small triangle has sides 3 and 5, and the matching sides in a larger triangle are 9 and x, then 3/5 = 9/x, which gives x = 15. Scale models and blueprints use the same idea, which is why proportions are a staple of geometry courses.