Proportion Calculator
Enter any three values and the fourth is solved instantly.
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- Proportion—
- Ratio as a decimal—
- Ratio as a percentage—
- Simplified ratio—
- Scale factor (left to right)—
- Cross products—
How to use this proportion calculator
- Type the three values you know into A, B, C or D. The layout matches the way a proportion is written: A over B equals C over D.
- Leave the unknown box empty. The calculator works out which one is missing and fills it in, highlighted in blue.
- Read the working underneath. It shows the cross multiplication line by line, so you can copy the method into your homework or check a figure you worked out by hand.
- Fill in all four boxes if you want to test whether two ratios are actually equal. The calculator compares the cross products and tells you whether the proportion holds.
Decimals and negative numbers are fine. The only value that will not work is a zero in a denominator position, because dividing by zero has no answer.
What is a proportion?
A ratio compares two quantities: three cups of flour to four cups of water is the ratio 3:4. A proportion is a statement that two ratios are equal. Write it as a fraction on each side of an equals sign and you have the standard form:
Read it out loud as “A is to B as C is to D”. If you scale a recipe, enlarge a photograph, read a map, convert currency or work out a unit price, you are solving a proportion whether you call it that or not. The relationship holds because both sides share the same underlying rate, so once three of the four numbers are known, the fourth is fixed.
Ratio, proportion and rate
- Ratio — a comparison of two quantities, such as 3:4.
- Rate — a ratio where the two quantities have different units, such as 60 miles per hour.
- Proportion — an equation stating that two ratios are equal, such as 3/4 = 15/20.
The proportion formula
Every proportion rests on one rule, known as cross multiplication or the rule of three. If two fractions are equal, then multiplying diagonally across the equals sign gives the same result both ways:
Rearrange that single identity and you get a formula for whichever value is missing:
| Unknown | Formula | Example |
|---|---|---|
| A | A = (B × C) / D | A = (4 × 15) / 20 = 3 |
| B | B = (A × D) / C | B = (3 × 20) / 15 = 4 |
| C | C = (A × D) / B | C = (3 × 20) / 4 = 15 |
| D | D = (B × C) / A | D = (4 × 15) / 3 = 20 |
Notice the pattern: the unknown always equals the product of the two values diagonally opposite and beside it, divided by the value directly across from it.
How to solve a proportion by hand
Take the proportion 3/4 = 15/x and work through it:
- Write it out. 3/4 = 15/x
- Cross multiply. Multiply each numerator by the opposite denominator: 3 × x = 4 × 15
- Simplify the known side. 3x = 60
- Isolate the unknown. Divide both sides by 3: x = 60 / 3
- Solve. x = 20
- Check it. 3/4 = 0.75 and 15/20 = 0.75. Both sides match, so the answer is right.
That final check is worth the ten seconds it takes. Converting both fractions to decimals catches almost every arithmetic slip, and it is exactly what the calculator above does in the background.
The shortcut when the numbers are friendly
Sometimes you can skip the algebra entirely. In 3/4 = 15/x, notice that 15 is 3 multiplied by 5. Whatever you do to the top you must do to the bottom, so x is 4 multiplied by 5, which is 20. This is called the scale factor method and it is faster than cross multiplication whenever one side divides neatly into the other.
Worked examples
Scaling a recipe
A bread recipe uses 2 cups of flour for 3 people. You are cooking for 18. How much flour?
Reading a map scale
A map is drawn at 1:25,000, meaning 1 cm on paper is 25,000 cm on the ground. Two towns are 6 cm apart on the map. How far apart are they in reality?
Comparing unit prices
A 4.5 kg bag of coffee costs £3 per kilo at the wholesaler. What would 7 kg cost at the same rate?
Direct and inverse proportion
The calculator above handles direct proportion, where the two quantities rise and fall together. There is a second kind worth knowing, because using the wrong one produces answers that look plausible and are completely wrong.
| Direct proportion | Inverse proportion | |
|---|---|---|
| Behaviour | One doubles, the other doubles | One doubles, the other halves |
| Equation | y = kx | y = k / x |
| Constant | y / x = k | x × y = k |
| Graph | Straight line through the origin | Curve, never touching either axis |
| Example | Hours worked and wages earned | Workers on a job and days to finish it |
The value k is the constant of proportionality. In a direct relationship it is the rate: pay someone £18 an hour and k is 18, whether they work two hours or forty.
Where proportions show up in tech and AI
Proportional reasoning is not just a school topic. It sits underneath a surprising amount of the work covered on this site:
- Aspect ratios. Resizing an image from 1920 pixels wide to 1280 without distortion is the proportion 1920/1080 = 1280/x, giving 720.
- Token estimates. The rough rule that one token is about 0.75 words is a ratio. Scaling it to a 128,000 token context window is a proportion.
- API costs. Pricing quoted per million tokens is a rate, and working out the bill for 37,000 tokens is the rule of three.
- Position sizing. Risking a fixed percentage of a portfolio on each trade keeps your stake proportional to your account as it grows and shrinks.
- Benchmark comparison. Frames per second on one GPU versus another at the same settings is a ratio, which is why results are quoted as a multiple rather than a raw number.
Common proportions worth memorising
| Ratio | As a decimal | As a percentage | Where you meet it |
|---|---|---|---|
| 1:2 | 0.5 | 50% | Half of anything |
| 2:3 | 0.667 | 66.7% | Classic page proportion |
| 3:4 | 0.75 | 75% | Older monitors, tablet cameras |
| 9:16 | 0.563 | 56.3% | Vertical video, phone screens |
| 16:9 | 1.778 | 177.8% | Widescreen video |
| 1:1.618 | 0.618 | 61.8% | The golden ratio |
The golden ratio deserves its own note. It is the only proportion where the ratio of the whole to the larger part equals the ratio of the larger part to the smaller, which is why it keeps reappearing in design, architecture and the arrangement of seeds in a sunflower head.
Frequently asked questions
What is the formula for a proportion?
The formula is A/B = C/D, solved by cross multiplication: A × D = B × C. Rearranging that gives you the missing value. If D is unknown, D = (B × C) / A.
How do I know if two ratios form a proportion?
Multiply diagonally. If A × D gives the same answer as B × C, the two ratios are equal and you have a true proportion. Converting both fractions to decimals works just as well and is often quicker in your head.
What is the difference between a ratio and a proportion?
A ratio compares two quantities, such as 3:4. A proportion is an equation stating that two ratios are equal, such as 3/4 = 15/20. A ratio is a single comparison; a proportion is a relationship between two of them.
Can a proportion have decimals or negative numbers?
Yes. Cross multiplication works the same way for decimals, fractions and negatives. The only restriction is that no denominator can be zero, since division by zero is undefined.
What is the constant of proportionality?
It is the fixed number linking two directly proportional quantities, written k in the equation y = kx. If you earn £18 for every hour worked, k is 18. In an inverse relationship the constant is the product instead: x × y = k.
What is the rule of three?
It is the traditional name for solving a proportion when three values are known and one is not. Multiply the two diagonally opposite known values, then divide by the third. It is the same arithmetic as cross multiplication, just a different name for it.
How do I scale a recipe using proportions?
Put the original quantity and the original number of servings on one side, and the unknown quantity over your new number of servings on the other. For 2 cups serving 3 people scaled to 18 people: 2/3 = x/18, so x = 12 cups.
Why does cross multiplication work?
Because multiplying both sides of an equation by the same thing keeps it true. Starting from A/B = C/D, multiply both sides by B and by D. The denominators cancel and you are left with A × D = B × C.