Ratio Calculator — Simplify Ratios & Solve Proportions
Simplify any ratio to lowest terms, solve proportions like 12:18 = ?:9 (the answer is 6, not 9), scale ratios for recipes and mixtures, and convert ratios to fractions and percents. For 12:18: divide both sides by their GCD 6 to get 2:3. Split 25 in the ratio 2:3 and you get 10 and 15 — every answer with the exact steps.
Ratio Calculator: Simplify, Proportions, Scaling & Conversions
Pick the mode that matches what you need. Calculations run in your browser.
What Is a Ratio Calculator?
A ratio calculator works with the ratio a:b — how two quantities compare. This free tool handles four jobs: simplify a ratio (12:18 becomes 2:3), find a missing term in a proportion (12:18 = ?:9 gives 6), scale a ratio up or down (2:3 scaled by 5 is 10:15), and convert a ratio to fractions and percents (2:3 is 2/5 and 3/5, or 40% and 60%). Every answer comes with the exact steps so you can verify it by hand.
How to Use the Ratio Calculator
The tool has four modes so you can start with whatever you need to find.
- Choose a mode that matches your problem (or tap a preset to load a worked example).
- Enter your values — for the Missing Term mode, leave exactly one of the four fields empty.
- Select Calculate to generate the result cards.
- Read the note under the cards for the step-by-step working and a check you can verify by hand.
The built-in check re-derives the answer a second way — multiplying back for simplification, matching cross-products for proportions, and summing to 100% for percent conversions.
What Is a Ratio?
A ratio compares two quantities part to part. The ratio 2:3 means: for every 2 parts of the first quantity, there are 3 parts of the second. The two sides always move together — you can multiply or divide both by the same number, but never just one.
Ratios wear three different outfits depending on the question: part-to-part (ratio), part-to-whole (fraction), or parts-per-hundred (percent). They all describe the same relationship, as the table shows.
| Form | What It Means | 2:3 Example |
|---|---|---|
| Ratio | Compares part to part | 2:3 — two parts of A for every three parts of B |
| Fraction | Compares each part to the whole | 2:3 -> 2/5 of the whole and 3/5 of the whole |
| Percent | Parts per hundred | 2:3 -> 40% of the whole and 60% of the whole |
Ratio Calculator Worked Examples
Every row below is computed exactly — hand-verified in Python — so you can check each one by hand.
| Task | Working | Answer |
|---|---|---|
| Simplify 12:18 | GCD(12,18) = 6; 12/6 = 2, 18/6 = 3 | 2:3 |
| Missing term 12:18 = ?:9 | ? = (12 x 9)/18 = 108/18 | 6 |
| Scale 2:3 by 5 | 2 x 5 = 10, 3 x 5 = 15 | 10:15 |
| Split 25 in ratio 2:3 | Parts = 5; each = 25/5 = 5; 2 x 5 = 10, 3 x 5 = 15 | 10 and 15 |
| 2:3 as percents | Whole = 5; 2/5 = 40%, 3/5 = 60% | 40% : 60% |
| 2:3 as fractions | Whole = 5; shares 2 and 3 | 2/5 and 3/5 |
The 12:18 = ?:9 Proportion Trap
This is the most-tested proportion trap in school math: for 12:18 = ?:9, students answer 9 because 9 is half of 18 — so the missing number "must" be half of 12... except that is not how proportions work either. Cross-multiplication says ? = (12 x 9)/18 = 108/18 = 6. Check it: 12 x 9 = 108 and 18 x 6 = 108 — the cross-products match. With ?=9 they would be 108 vs 162 — a fail. Set the tool to Missing Term, load the 12:18 = ?:9 preset, and watch the trap dissolve into one equation.
Splitting a Whole: 2:3 of 25
To split 25 in the ratio 2:3, add the sides: 2 + 3 = 5 total parts. Each part is worth 25/5 = 5. The first share gets 2 x 5 = 10 and the second gets 3 x 5 = 15. The check is instant: 10 + 15 = 25. This is the recipe-scaling move — flour to sugar 2:3 with 25 cups total means 10 cups of flour and 15 of sugar.
How to Simplify a Ratio Step by Step
Follow this recipe for the classic example — simplifying 12:18.
- Find the GCD: list the divisors of 12 (1, 2, 3, 4, 6, 12) and 18 (1, 2, 3, 6, 9, 18). The greatest common one is 6.
- Divide the first term: 12/6 = 2.
- Divide the second term: 18/6 = 3.
- Write the simplified ratio: 2:3.
- Check: 2 x 6 = 12 and 3 x 6 = 18 — multiplying back restores the original.
Common Mistakes With Ratios
1. Stopping before the GCD
12:18 divided by 2 gives 6:9 — but that is not finished. Until both sides share no common factor except 1, keep dividing: 6:9 divided by 3 gives 2:3. "Can both sides still divide by something?" is the question to ask after every step.
2. Falling for the proportion trap
In 12:18 = ?:9, answering 9 because "9 is half of 18" is the classic fail. The missing term is 6 — cross-multiply ((12 x 9)/18 = 6) and check the cross-products (108 = 108) instead of guessing from patterns.
3. Cross-multiplying the wrong pairs
In a:b = c:d the cross pairs are a x d and b x c — the diagonals. Multiplying a x b (the two terms of the first ratio) answers a different question entirely. Trace an X across the proportion before you multiply.
4. Confusing a:b with the fraction a/b
2:3 is NOT the fraction 2/3. A ratio compares part to part; the fraction of the whole is a/(a+b): 2:3 gives 2/5 and 3/5. Writing 2/3 means "2 parts out of 3" — the ratio 2:1, a completely different mixture.
5. Scaling only one side
Scaling 2:3 by 5 means 10:15 — both sides. Changing only one side (2:3 -> 10:3) breaks the ratio: it now simplifies to 10:3, not 2:3. Recipes go wrong exactly this way; the tool's Scale mode multiplies both sides at once.
6. Turning 2:3 into "2% and 3%"
Percents measure parts per hundred of the whole, and the whole is a + b = 5. So 2:3 is 2/5 = 40% and 3/5 = 60% — never 2% and 3%. The sanity check: the percents must add to 100%.
Benefits and Limitations
Benefits
- Checks homework instantly: simplify 12:18 or solve 12:18 = ?:9 with every step shown.
- The built-in cross-product check catches the ?=9 trap answer on sight.
- Scales recipes and mixtures up or down without the one-sided scaling error.
- Fraction and percent conversions always sum correctly (fractions to 1, percents to 100%).
- Plain-ASCII math notation renders cleanly in every browser and font.
- Responsive on phones, tablets and desktops.
Limitations
- Works with two-term ratios (a:b) only — compound ratios like 2:3:4 are not supported.
- Decimal answers are shown to four places; exact fractions appear where the math is exact.
- Very large values (above ~10^15) can lose precision in browser floating point.
Quick Definitions
| Question | Direct Answer |
|---|---|
| What is a ratio? | A part-to-part comparison of two quantities, written a:b. 12:18 means 12 parts of the first for every 18 of the second; in lowest terms it is 2:3. |
| What is a proportion? | An equation stating two ratios are equal: a:b = c:d. 12:18 = 6:9 is a proportion because both simplify to 2:3. |
| What is the GCD? | The greatest common divisor — the largest number dividing both terms. GCD(12,18) = 6, and dividing both sides by it simplifies 12:18 to 2:3. |
| What is cross-multiplication? | The proportion rule a:b = c:d -> a x d = b x c. For 12:18 = ?:9, ? = (12 x 9)/18 = 6. |
| What does it mean to scale a ratio? | Multiplying both sides by the same number k. 2:3 scaled by 5 is 10:15; the ratio is unchanged. |
| How is a ratio different from a fraction? | A ratio compares part to part (2:3); a fraction compares part to whole. 2:3 gives the fractions 2/5 and 3/5 of the whole. |
| What is a unit rate? | A ratio with a second term of 1, e.g. 2:3 = 1:1.5 — useful for prices ("$1.50 per item") and speeds ("60 miles per hour"). |
| What is an equivalent ratio? | A ratio with the same lowest terms: 12:18, 6:9, 4:6 and 10:15 are all equivalent to 2:3. |
Frequently Asked Questions
How do you simplify the ratio 12:18?
Divide both sides by their greatest common divisor (GCD). GCD(12,18) = 6, so 12/6 = 2 and 18/6 = 3: the ratio simplifies to 2:3. Check it by multiplying back: 2 x 6 = 12 and 3 x 6 = 18.
What is the missing term in 12:18 = ?:9?
6. Cross-multiply: ? = (12 x 9)/18 = 108/18 = 6 (check: 12 x 9 = 108 and 18 x 6 = 108). Nine is the classic trap — it feels right because 9 is half of 18, but 12 x 9 = 108 while 18 x 9 = 162, so the proportion fails.
How do you split 25 in the ratio 2:3?
The shares are 10 and 15. Total parts = 2 + 3 = 5, each part is worth 25/5 = 5, so the shares are 2 x 5 = 10 and 3 x 5 = 15 (check: 10 + 15 = 25).
How do you convert the ratio 2:3 to a percent?
40% : 60%. Add the sides to find the whole: 2 + 3 = 5. Then 2/5 = 40% and 3/5 = 60% (check: 40% + 60% = 100%). The classic mistake is treating it as 2/3 = 66.7% — but a ratio is part-to-part, so the whole is 5, not 3.
Is 2:3 the same as the fraction 2/3?
No. A ratio compares part to part: 2:3 splits a whole into 5 parts, giving the fractions 2/5 and 3/5. The fraction 2/3 means 2 parts out of 3 — which is the ratio 2:1, not 2:3.
How do you scale a ratio?
Multiply both sides by the same number. 2:3 scaled by 5 is 10:15 (2 x 5 = 10, 3 x 5 = 15), which simplifies straight back to 2:3. Scaling only one side breaks the ratio.
Related Math Tools
Ratio Calculator Methodology
This calculator derives every answer from the four standard ratio relations, using exact decimal arithmetic and the Euclidean algorithm for the GCD.
GCD(12,18) is computed by the Euclidean algorithm: GCD(12,18) = GCD(18,12) = GCD(12,6) = GCD(6,0) = 6. The Missing Term mode accepts exactly one empty field; division by a zero divisor is caught before any calculation. The page is intended as an educational calculator and verification tool.
Mathematical References
Find Your Ratio
Enter a ratio above — simplify it to lowest terms, find a missing term, scale it for a recipe, or convert it to fractions and percents with the exact steps shown.
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