The short answer
A ratio compares two quantities by division, written a:b. To simplify, divide both sides by their GCF — 12:8 becomes 3:2. To solve a proportion (two equal ratios, a/b = c/d), cross-multiply: ad = bc, then solve for the unknown. To split a total according to a ratio, divide the total by the number of parts, then multiply by each ratio value.
Key takeaways
- Simplifying a ratio uses the exact same GCF logic as simplifying a fraction.
- A proportion is just two equivalent ratios set equal to each other; cross-multiplication is the standard way to solve for a missing value.
- Splitting a total by ratio always starts with dividing by the sum of the parts, not the total itself.
- A ratio (part-to-part, like boys to girls) and a fraction (part-to-whole, like boys out of the whole class) describe related but different things from the same numbers.
Simplifying a ratio
12:8 — GCF(12, 8) = 4 → 12÷4 : 8÷4 = 3:2
A ratio is fully simplified once its two numbers share no common factor besides 1 — exactly the same stopping point used when simplifying a fraction to lowest terms.
Solving a proportion
3/4 = x/12
3 × 12 = 4 × x → 36 = 4x → x = 9
Cross-multiplication works because a proportion states two fractions are equal, and multiplying both sides of that equation by both denominators clears the fractions entirely, leaving a simple equation to solve.
Worked example: splitting a total by ratio
Split $100 between two people in a 2:3 ratio.
Total parts = 2 + 3 = 5
Value of one part = $100 ÷ 5 = $20
Split: 2 × $20 = $40 and 3 × $20 = $60
Check: $40 + $60 = $100, and 40:60 simplifies back down to 2:3 — confirming the split matches the original ratio exactly.
Common mistakes to avoid
- Dividing the total by one of the ratio numbers instead of by the sum of all the parts.
- Cross-multiplying the wrong pairs when solving a proportion — it's always numerator × opposite denominator on each side.
- Ignoring order in a ratio — 3:2 and 2:3 describe different splits, so which quantity comes first genuinely matters.
- Treating a ratio as a percentage without converting it — 3:2 isn't "3% to 2%"; each side needs to be divided by the total and multiplied by 100 first.
Related calculators
- Fraction Calculator — apply the same simplification logic to fractions.
- Percentage Calculator — convert a ratio's parts into percentages of the whole.
- Greatest Common Factor Calculator — double-check the GCF used to simplify a ratio.
- Number Sequence Calculator — see ratios at work as the common ratio in a geometric sequence.