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This page reviews the key ideas behind ratios and proportions. Whether you're a student brushing up before a test, or a parent helping with homework, these reminders should get you back on track quickly.
What is a Ratio?
A ratio is a way of comparing two quantities. For example, if a recipe calls for 2 cups of flour and 3 cups of oats, the ratio of flour to oats is 2 to 3.
Ratios can be written three ways, and they all mean the same thing:
- Using a fraction: 2/3
- Using a colon: 2:3
- Using words: 2 to 3
The order matters — 2:3 and 3:2 are different ratios, just as 2/3 and 3/2 are different fractions.
Equivalent Ratios
Two ratios are equivalent if they describe the same relationship, just scaled up or down. This works exactly like equivalent fractions: multiply or divide both numbers by the same value.
Example: Fill in the blank: 3/4 = ___/12
Ask yourself: what did the denominator 4 get multiplied by to reach 12?
4 × 3 = 12 → multiply the numerator by the same number
3 × 3 = 9
So: 3/4 = 9/12
The same idea works in colon notation: 3:4 = 9:12.
You can also work backwards — if the numbers on the right are smaller than on the left, divide instead of multiply.
Solving for X (Cross-Multiplication)
When the missing value isn’t as easy to spot, use cross-multiplication. This is the standard method for solving proportion problems of the form:
a/b = X/d or a/b = c/X
The rule: multiply diagonally across the equals sign, then solve.
Example 1: 3/5 = X/25
Cross-multiply: 3 × 25 = 5 × X
75 = 5X
Divide both sides by 5: X = 75 ÷ 5 = 15
Answer: 3/5 = 15/25
Example 2: 3/5 = 15/X
Cross-multiply: 3 × X = 5 × 15
3X = 75
Divide both sides by 3: X = 75 ÷ 3 = 25
Answer: 3/5 = 15/25
Why Does Cross-Multiplication Work?
Cross-multiplication is really just a shortcut for multiplying both sides of the equation by both denominators at once. Starting with:
3/5 = X/25
Multiply both sides by 5 × 25 = 125:
(3/5) × 125 = (X/25) × 125
3 × 25 = X × 5
75 = 5X → X = 15
The denominators cancel out, leaving exactly what cross-multiplication gives you.
Checking Your Answer
Once you find X, it’s easy to verify your answer — just substitute it back in and confirm both sides of the proportion are equal.
Example: We found that 3/5 = 15/25. To check:
Divide both fractions down to lowest terms:
3/5 is already in lowest terms.
15/25: both divisible by 5 → 15÷5 = 3, 25÷5 = 5 → 3/5 ✓
Both sides equal 3/5, so the answer is correct.
Alternatively, use cross-multiplication to check: if 3 × 25 = 5 × 15, then 75 = 75. ✓
← Back to the Ratios and Proportions Worksheet Creator
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