What a proportion says
A proportion is a statement that two ratios are equal. 3/4 = 9/12 says that 3 compares to 4 in exactly the same way 9 compares to 12. When one of the four numbers is missing, the proportion becomes a puzzle: what number keeps the two ratios equal?
You'll see them in recipes, map scales, unit prices, similar triangles and percent problems. The math is the same in every case. The hard part is usually putting the numbers in the right places.
Step 1: set it up with labels
Write the units next to every number before you do any math. Say 5 pounds of apples cost $8 and you want the cost of 12 pounds. Put dollars on top and pounds on the bottom on both sides: $8 / 5 lb = $x / 12 lb.
You could also put pounds on top on both sides. Either way works as long as both sides follow the same pattern. Flipping just one side is the classic mistake, and it gives an answer that looks plausible but is wrong.
Step 2: cross multiply
Multiply the top of each fraction by the bottom of the other. For 8/5 = x/12, that's 8 x 12 = 5 x x, or 5x = 96. This works because multiplying both sides of the equation by 5 and by 12 clears both fractions at once.
Then divide by the number stuck to x. 96 / 5 = 19.2, so 12 pounds cost $19.20. Check by plugging it back in: 8 / 5 = 1.6 and 19.2 / 12 = 1.6. Same ratio, so the answer holds.
Formula notes
a / b = c / d means a x d = b x c
A faster route: the unit rate
Sometimes it's quicker to find the value of one unit first. $8 for 5 pounds is $8 / 5 = $1.60 per pound. Twelve pounds is then 12 x $1.60 = $19.20. You get the same answer as cross multiplying, and the unit rate is often useful on its own.
Scale factors work too. If a recipe uses 2 cups of flour for 12 cookies and you want 30 cookies, that's 30 / 12 = 2.5 times the recipe, so 5 cups of flour. Pick whichever route has the friendlier numbers.
When a problem isn't proportional
Cross multiplication only works if the relationship is proportional, meaning doubling one amount doubles the other. Plenty of everyday situations break that rule.
A taxi with a $3 base fare plus $2 per mile costs $13 for 5 miles but $23 for 10. Doubling the distance didn't double the price, so a proportion would give the wrong fare. The same goes for bulk discounts, tax brackets and anything with a fixed fee.
Some problems are proportional the other way round. More workers on the same job means fewer hours, so the product of workers and hours stays fixed instead of their ratio. That's inverse proportion, and you solve it by setting the products equal: 6 x 40 = 10 x h gives h = 24.
A quick checklist
Run through these before you trust an answer. When you just want the number with the check done for you, our proportion calculator solves for any blank and shows both cross products.
- Same units on both sides, and same quantity type in matching positions.
- No zero in a denominator.
- The relationship is truly proportional, with no fixed fees or caps.
- The answer is the right size. More pounds should cost more.
- Both cross products match when you plug the answer back in.
Frequently asked questions
Why does cross multiplication work?
It's a shortcut for multiplying both sides of the equation by both denominators. That clears the fractions and leaves a plain equation you can solve.
Can a proportion have decimals or fractions in it?
Yes. Cross multiply as usual. 0.5/2 = x/10 gives 2x = 5, so x = 2.5.
How do you solve a proportion with x in two places?
Cross multiply and collect the x terms on one side. For x/4 = (x + 3)/10, you get 10x = 4x + 12, so 6x = 12 and x = 2.
What's the easiest way to check a proportion answer?
Divide top by bottom on each side. If both give the same decimal, the proportion is true.
References
These sources support the method or guidance used for Proportion Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Proportion Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.