Proportion Calculator: Solve a/b = c/d for Any Missing Value
A proportion is a statement that two ratios are equal: a/b = c/d. Given any three of the four values, the fourth can be found in one step. This comes up constantly — scaling recipes, reading maps, converting units, resizing images — but the arithmetic is easy to fumble under pressure.
Cross-Multiplication: The One Rule You Need
If a/b = c/d, then a × d = b × c. The cross products are equal. This gives you a formula for each missing value:
- Solve for d: d = b × c / a
- Solve for c: c = a × d / b
- Solve for b: b = a × d / c
- Solve for a: a = b × c / d
All four formulas are the same operation rearranged. Cross-multiply to get the numerator, divide by the remaining known value.
Why This Works
Start with a/b = c/d. Multiply both sides by b × d:
a × d = b × c
That's it. The equality of ratios forces the equality of cross products. If you ever get a × d ≠ b × c, the original statement a/b = c/d was false — the ratios were not equal.
Real-World Examples
Recipe scaling
A recipe serves 4 and calls for 3 cups of flour. How much flour for 10 servings?
3/4 = ?/10 → ? = 3 × 10 / 4 = 7.5 cups
Map distance
A map scale is 1 cm = 25 km. Two cities are 6.4 cm apart on the map.
1/25 = 6.4/? → ? = 25 × 6.4 / 1 = 160 km
Image resizing
An image is 800 × 600 px. Resize to 400px wide, keeping proportions.
800/600 = 400/? → ? = 600 × 400 / 800 = 300 px
Unit conversion
If 1 mile = 1.60934 km, how many km is 13.1 miles (a half marathon)?
1/1.60934 = 13.1/? → ? = 1.60934 × 13.1 = 21.08 km
Mixing paint
A custom color uses 3 parts red to 5 parts white. You need 2 liters of white. How much red?
3/5 = ?/2 → ? = 3 × 2 / 5 = 1.2 liters
When Proportions Break
Proportions assume a linear relationship — double one variable, double the other. This holds for most scaling problems but fails in several common situations:
| Scenario | Problem |
|---|---|
| Cooking time for a larger roast | Heat penetration is not linear — a 10 lb roast doesn't take exactly twice as long as a 5 lb one |
| Area vs side length | Doubling the side of a square quadruples the area — use the square of the ratio |
| Volume vs side length | Doubling the side of a cube multiplies volume by 8 |
| Mortgage payments | Interest compounds — a loan twice as large doesn't have exactly twice the payment |
For linear relationships, cross-multiplication is exact. For non-linear ones, it gives only an approximation.
Checking a Proportion
If you have all four values and want to verify they form a proportion, check whether a × d = b × c. The calculator's verification mode does this and shows the deviation — useful when working with measured values that have rounding error.
Frequently Asked Questions
Can I use a proportion calculator to adjust cooking time for a larger roast?
No — cooking time doesn't scale proportionally with weight. A 10 lb roast won't take exactly twice as long as a 5 lb one because heat penetration into meat is non-linear. The proportion calculator works only for true linear relationships like recipe ingredient scaling, map distances, and unit conversions, so you'd need to consult cooking guidelines based on weight and temperature instead.
What does it mean when a × d ≠ b × c in a proportion?
It means the ratios are not equal — the values don't form a valid proportion. For example, if you're checking whether 3/4 = 6/9, cross-multiplying gives 3 × 9 = 27 and 4 × 6 = 24, which aren't equal, so the proportion is false. This verification check helps catch errors when you have all four values.
How do I resize an image to 400 pixels wide while keeping the same proportions?
Set up the proportion as original_width/original_height = 400/unknown_height. If your image is 800 × 600 px, enter 800/600 = 400/? and solve for the missing value. The calculator gives you 300 px, so your resized image will be 400 × 300 pixels with the same aspect ratio.
Why does doubling the side of a square not work with the proportion calculator?
Because area scales with the square of the side length, not linearly. If you double a square's side, the area quadruples — not doubles. The proportion calculator assumes a/b = c/d (linear relationships), so it's not designed for geometric scaling problems involving area or volume, which follow non-linear rules.
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