Ratio & proportion tools
What Is a Ratio?
A ratio compares quantities by how many parts each one gets. It fixes the relationship, not the amount.
3:2 says that for every three of the first quantity there are two of the second. It does not say there are five of anything. The same ratio describes 3 and 2, 30 and 20, 4.5 and 3, or 300 kg and 200 kg — every pair where the first is one and a half times the second.
That is the whole idea, and almost every other ratio question is a consequence of it: because only the relationship is fixed, you can rescale freely; because the parts add up, you can also ask what share each one holds of the total.
Part to part, and part to whole
The two readings live in the same picture. Along the bar, the parts compare with each other. Bracketed against the full bar, each part compares with the total.
Confusing the two is the most common source of wrong ratio answers, and it is easy to detect: part-to-whole figures must sum to 100%, part-to-part figures need not. 60% and 40% are shares of the whole. 150% is the first part measured against the second. Both are correct answers to different questions about 3:2.
| Question | Comparison | For 3:2 |
|---|---|---|
| How does the first compare with the second? | Part to part | 3 ÷ 2 = 1.5, or 150% |
| How does the second compare with the first? | Part to part, reversed | 2 ÷ 3 = 0.667, or 66.7% |
| What share of everything is the first? | Part to whole | 3/5 = 60% |
| What share of everything is the second? | Part to whole | 2/5 = 40% |
| How many parts are there in total? | Sum of parts | 3 + 2 = 5 |
Only the last two rows add to 100%. If your two percentages do not, you are looking at a part-to-part answer.
Order is part of the statement
3:2 and 2:3 are different ratios: the first quantity is larger in one and smaller in the other. Ratios are usually written in the order the labels were written, so “three parts concentrate to two parts water” is 3:2 and never 2:3. The only ratio that survives reversal unchanged is 1:1.
Notation
The colon form 3:2 and the fraction form 3/2 carry the same information for two quantities, but they invite different mistakes. 3/2 looks like a part-to-whole fraction and is not — the whole here is 5. Colon notation also extends to three or more quantities, where a single fraction cannot: 2:3:5 is a legitimate ratio, and it is not one number.
Units come before the ratio
A ratio is a pure number, so the units have to be reconciled before you write it down. 500 g compared with 1 kg is not 500:1.
500 g and 1 kg
Convert to a common unit: 1 kg = 1000 g.
Write the comparison in that unit: 500:1000.
Divide both parts by 500: the ratio is 1:2.
Read it back: the first quantity is half the second, which matches half a kilo against a kilo.
Calculating a ratio from two quantities
Method
- Decide which quantity is written first — usually the order in your source.
- Put both quantities in the same unit.
- Write them as A:B.
- Divide both parts by their greatest common divisor.
- Multiply the result back up to confirm you return to the original numbers.
12 and 18
Same unit, order given, so the comparison is 12:18.
The greatest common divisor of 12 and 18 is 6.
12 ÷ 6 = 2 and 18 ÷ 6 = 3, so the ratio is 2:3.
Check: 2 × 6 = 12 and 3 × 6 = 18, the same multiplier both times.
Total parts 5, so the shares are 40% and 60%.
The check has diagnostic value. If multiplying back returns one original number but not the other, the division was uneven; if both are off by a consistent amount, the unit conversion was the problem.
Equivalent ratios
Multiply or divide every part by the same non-zero number and the comparison is untouched, because (k × 3)/(k × 2) = 3/2 for any k. 3:2, 6:4 and 30:20 are the same ratio written at three sizes. Simplifying runs that rule downward; scaling a recipe runs it upward. Adding to both parts is a different operation entirely and does not preserve the ratio.
More than two parts
2:3:5 works the same way: ten parts in total, shares of 20%, 30% and 50%. Any two of the three still form an ordinary two-part ratio, so 2:3:5 also tells you the first and second stand in 2:3. What you lose is the single-fraction shorthand — there is no one number that captures a three-way comparison.