How to Solve Proportions, Step by Step
A proportion says two ratios are equal. Once you can spot that shape in a question, scaling recipes, reading map distances, and unit pricing all become the same three-line problem.
What a proportion actually is
A ratio compares two quantities. A proportion says two ratios are equal, written a/b = c/d. Solving one means finding the single missing value that keeps them equal.
The method
- Write the two ratios as fractions, with the same kind of quantity on top in both.
- Cross multiply: a/b = c/d becomes a x d = b x c.
- Solve the resulting one-step equation.
- Check by putting the value back and comparing the two fractions.
Why the units have to match
The single thing that makes proportions go wrong is putting cups on top in one fraction and cookies on top in the other. Decide what goes on top, write both fractions the same way, and the cross multiplication takes care of itself.
Worked examples
Example 1
Question
Solve for x: 3/4 = x/20
Answer
x = 15
Steps
- 1Cross multiply: 3 x 20 = 4 x x.
- 2That gives 60 = 4x.
- 3Divide both sides by 4: x = 15.
- 4Check: 15/20 simplifies to 3/4. The two ratios match.
Example 2
Question
2 cups of flour make 12 cookies. How much flour makes 30 cookies?
Answer
5 cups
Steps
- 1Put flour on top in both fractions: 2/12 = x/30.
- 2Cross multiply: 2 x 30 = 12 x x, so 60 = 12x.
- 3Divide by 12: x = 5 cups.
- 4Sense check: 30 cookies is two and a half times 12, and 5 is two and a half times 2.
Writing 12/2 = 30/x works too, as long as both fractions have cookies on top. Mixing the two orders is what produces a wrong answer.
Example 3
Question
A map uses a scale of 1 cm to 25 km. How far apart are two towns 4.5 cm apart on the map?
Answer
112.5 km
Steps
- 1Set up with map distance on top: 1/25 = 4.5/x.
- 2Cross multiply: 1 x x = 25 x 4.5.
- 3So x = 112.5 km.
- 4Sense check: 4.5 times as far on the map means 4.5 times as far in reality, and 4.5 x 25 = 112.5.
Common mistakes
Writing 2/12 = 30/x when the first fraction has flour on top and the second has cookies on top.
Keep the same quantity on top in both fractions. Label them if you have to: flour/cookies = flour/cookies.
Adding instead of scaling: 12 cookies needs 2 cups, so 30 cookies needs 2 + 18.
Proportions multiply. If the output is 2.5 times bigger, every ingredient is 2.5 times bigger.
Cross multiplying and then forgetting to divide, leaving 60 = 4x as the answer.
Finish the equation. The answer is the value of x, not the line before it.
Mixing units, like minutes on one side and hours on the other.
Convert to the same unit before setting up the proportion, not after.
Practice it
Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.
1. Solve: 5/8 = x/24
Show answer
x = 15. Cross multiply: 5 x 24 = 8x, so 120 = 8x.
2. Solve: 7/x = 21/9
Show answer
x = 3. Cross multiply: 7 x 9 = 21x, so 63 = 21x.
3. 9 pens cost $12. What do 15 pens cost at the same rate?
Show answer
$20. Set up 9/12 = 15/x, cross multiply to get 9x = 180.
Questions people ask
- Is cross multiplying always allowed?
- For a proportion, yes, as long as no denominator is zero. It is just multiplying both sides by both denominators.
- Does it matter which ratio I write first?
- No. a/b = c/d and c/d = a/b give the same cross multiplication. What matters is that the two fractions are built the same way.
- What if the unknown is on the bottom?
- Nothing changes. Cross multiply as usual and you will end up with the unknown multiplied by a number, which you then divide out.
- How do I know a question is a proportion?
- Look for a rate that stays constant: per cup, per hour, per centimetre, at the same rate. That phrase is the signal.
Got one in front of you?
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