Examples · Fractions
Ratio Examples: 7 Worked Problems
Seven ratios, ordered so each one adds exactly one new idea to the last — every answer here is checked by adding the parts back or reducing them again.
Simplifying a two-term ratio
- 12:18the ratio
- greatest common factor of 12 and 18 is 6the largest number dividing both evenly
- 12:18 → 2:3divide every term by 6
- 20:5the ratio
- greatest common factor of 20 and 5 is 55 divides into both
- 20:5 → 4:1divide every term by 5
Example 2 is worth noting: a ratio can simplify to end in 1, meaning the first quantity is exactly 4 times the second — a valid, fully simplified ratio, not an unfinished one. Both examples here compare a quantity part to part, directly against each other, rather than against a combined whole.
A three-term ratio
- 5:15:25a three-term ratio
- greatest common factor of 5, 15 and 25 is 55 divides evenly into all three
- 5:15:25 → 1:3:5divide every term by 5
The method does not change with a third term — find one factor that divides every term, and divide all of them by it together.
A ratio with decimals
- 0.6:0.9the ratio, with one decimal place each
- 0.6:0.9 → 6:9multiply every term by 10 to clear the decimals
- 6:9 → 2:3divide every term by their greatest common factor, 3
Splitting a total between two shares
- 9:12 → 3:4simplify first — fewer, smaller parts to work with
- 3 + 4 = 7the total number of parts
- 84 ÷ 7 = 12the value of one part
- 3 × 12 = 36, and 4 × 12 = 48each share gets its number of parts
- 36 + 48 = 84check: the parts add back to the total
Simplifying before splitting is not required — 9:12 and 3:4 give the identical split — but smaller numbers are easier to multiply by hand.
Splitting a total between three shares
- 1 + 2 + 3 = 6the total number of parts
- 120 ÷ 6 = 20the value of one part
- 1 × 20 = 20, 2 × 20 = 40, 3 × 20 = 60each share gets its number of parts
- 20 + 40 + 60 = 120check: the parts add back to the total
- 3 + 4 + 5 = 12the total number of parts
- 60 ÷ 12 = 5the value of one part
- 3 × 5 = 15, 4 × 5 = 20, 5 × 5 = 25each share gets its number of parts
- 15 + 20 + 25 = 60check: the parts add back to the total
Both examples use the same two-step method regardless of how many shares are involved: find the value of one part, then multiply every term by it.
Checking every result the same way
Two checks apply to every example above, and running both is worth making a habit rather than a step to skip when the answer looks right.
| Example | Check the answer | Result |
|---|---|---|
| 1 | 2 × 6 = 12, 3 × 6 = 18 | reduced ratio scales back to the original |
| 5 | 36 + 48 | 84 ✓ |
| 6 | 20 + 40 + 60 | 120 ✓ |
This is exactly what the Ratio Calculator on this site checks internally — a simplified ratio always scales back to the original by the factor divided out, and a split's parts always add back to the total exactly.
What these seven worked examples have in common
Every one of these worked examples reduces to the same two ideas: finding an equivalent ratio in lowest terms, or spreading a total across shares that stay proportional to a given ratio. A three-term ratio and a decimal ratio are not separate techniques — they are the identical step by step method, just applied to a slightly different starting shape.
The link back to fractions is worth keeping in mind throughout: a two-term ratio and a proportion share the same underlying arithmetic, which is exactly why a ratio calculator and a proportion calculator both lean on the greatest common factor and cross-multiplication.
Questions about ratio examples
What is the ratio 12:18 simplified?
2:3. Divide both terms by their greatest common factor, 6.
How do I split 84 in the ratio 9:12?
Simplify to 3:4 first, add the terms for 7 total parts, divide 84 by 7 to get 12 per part, then multiply: 36 and 48.
How do I simplify a ratio with decimals?
Multiply every term by the same power of 10 to clear the decimals, then simplify as usual. 0.6:0.9 becomes 6:9, which reduces to 2:3.
Can a simplified ratio end in 1?
Yes — 20:5 simplifies to 4:1, meaning the first quantity is exactly four times the second. That is a fully simplified ratio, not an incomplete one.