Quick answer
To simplify a ratio a:b, divide both terms by their GCD. 12:8 → GCD = 4 → 3:2. Equivalent ratios are found by multiplying both terms by the same number.
GCD = Greatest Common Divisor - the largest number that divides evenly into both terms. For 12 and 8, GCD = 4 because 4 is the highest factor shared by both.
How to use this ratio calculator
Enter the two terms of your ratio. Optionally enter a scale multiplier to generate a scaled equivalent. The calculator returns the simplified ratio, fraction form, decimal value, and step-by-step GCD workings.
What is a ratio?
A ratio compares two quantities and shows their relative sizes:
$$a : b = \frac{a}{b}$$
A ratio of 3:2 means for every 3 units of the first quantity, there are 2 units of the second. Ratios can represent parts-to-parts (mixing ratios) or parts-to-whole (share ratios). They are dimensionless - 3:2 is the same whether counting apples, millilitres, or dollars.
Simplifying ratios
A ratio is in its simplest form when the two terms share no common factor other than 1. To simplify, divide both by their GCD:
$$a : b = \frac{a}{\gcd(a,b)} : \frac{b}{\gcd(a,b)}$$
$$12 : 8 = \frac{12}{\gcd(12,8)} : \frac{8}{\gcd(12,8)} = \frac{12}{4} : \frac{8}{4} = 3 : 2$$
GCD(12,8) = 4. 12 ÷ 4 = 3, 8 ÷ 4 = 2. Simplified ratio: 3:2.
| Ratio | GCD | Simplified | As fraction |
|---|---|---|---|
| 4:6 | 2 | 2:3 | 2/3 |
| 15:10 | 5 | 3:2 | 3/2 |
| 18:24 | 6 | 3:4 | 3/4 |
| 100:75 | 25 | 4:3 | 4/3 |
| 7:14 | 7 | 1:2 | 1/2 |
Scaling ratios
Multiplying both terms by the same number k gives an equivalent ratio:
$$a : b = (a \times k) : (b \times k) \quad \text{for any } k > 0$$
To verify equivalence, cross-multiply:
$$a : b = c : d \iff a \times d = b \times c$$
3:2 = 15:10? 3 × 10 = 30 and 2 × 15 = 30. Equal products - equivalent. ✓
Scaling up is used in cooking (scaling a recipe), engineering (scale drawings), and finance (scaling returns to a portfolio size).
Dividing a quantity in a ratio
To divide a total amount T between two parts in ratio a:b:
$$\text{Part} = \frac{a}{a+b} \times \text{Total}$$
Divide £500 in ratio 3:2: total parts = 5. Part A = (3/5) × 500 = £300. Part B = (2/5) × 500 = £200. Check: 300 + 200 = 500 ✓.
Worked examples for ratio calculation
Example 1: simplify 36:48
GCD(36, 48) = 12. 36 ÷ 12 = 3, 48 ÷ 12 = 4. Simplified: 3:4.
Example 2: find a ratio equivalent to 5:3 with first term 20
Scale factor: 20 ÷ 5 = 4. Second term: 3 × 4 = 12. Equivalent ratio: 20:12.
Example 3: divide 90 kg in ratio 2:1
Total parts = 3. Part 1 = (2/3) × 90 = 60 kg. Part 2 = (1/3) × 90 = 30 kg.
Ratio tool suite
This hub covers ratio simplification and scaling. For more specific ratio calculations:
- Proportion Calculator - solve a:b = c:d for the missing value (cross-multiplication)
- Unit Rate Calculator - find the rate per single unit (e.g. price per item)
- Golden Ratio Calculator - find dimensions based on φ ≈ 1.618
Common mistakes when calculating ratios
Adding instead of multiplying to scale
3:2 + 4 ≠ 7:6. Adding 4 to both terms changes the ratio. To scale, multiply: 3:2 × 4 = 12:8.
Confusing ratio with fraction
A ratio 3:5 means 3 parts first and 5 parts second. As a fraction of the total, the first part is 3/(3+5) = 3/8 - not 3/5. The fraction a/b represents the ratio a:b, but the fraction of the whole is a/(a+b).
A partnership agreement specified profits would be split in a 2:3 ratio. The partner with the smaller share assumed they would receive 2/3 of all profits - reading the ratio directly as a fraction. The actual share is 2/(2+3) = 2/5 = 40%, not 66.7%.
The discrepancy was not caught until the first annual distribution. Ratio a:b gives shares of a/(a+b) and b/(a+b) of the total - the denominator is the sum of both terms, not just the other term.
A baker scaling a glaze recipe that called for cornstarch to water in a 1:4 ratio added 3 to each term to make a larger batch: 4:7. The resulting glaze came out noticeably thinner than the original.
Adding a constant to both terms does not preserve a ratio - 4/7 is 57% water, while 1/4 is 80% water. The correct approach is to multiply both terms by the same factor: 1:4 scaled by 3 gives 3:12, which holds the same proportion.
Frequently asked questions about ratios
How do you simplify a ratio?
Divide both terms by their GCD. 12:8 → GCD = 4 → 3:2.
What is a ratio?
A comparison of two quantities showing their relative sizes. Written a:b or a/b. Dimensionless.
How do you find an equivalent ratio?
Multiply both terms by the same number k. Verify with cross-multiplication.
Quiz: how well do you know ratios?
1. What does it mean for a ratio to be "dimensionless"?
2. What is the simplified form of the ratio 100:75?
3. Dividing £500 in ratio 3:2, how much does the larger share receive?
4. You want to scale ratio 5:3 so the first term becomes 20. What is the second term?
5. A ratio is written as 3:5. What fraction of the total does the first part represent?