Math · Live
Ratio Calculator,
simplify, solve, split.
Three tools in one: simplify any ratio to its lowest terms, solve a proportion (A:B = C:D) for any missing value using cross-multiplication, or divide a quantity into parts in a given ratio, all with step-by-step working.
Inputs
Configure ratio
Reduce A:B to lowest terms
Multiply the simplified ratio by this factor.
Equivalent ratios
Simplified ratio
- Decimal
- 1.5
- Fraction
- 3/2
- GCD
- 4
- As %
- 60% : 40%
Working
- Original ratio: 12 : 8
- GCD(12, 8) = 4
- 12 ÷ 4 = 3
- 8 ÷ 4 = 2
- Simplified ratio: 3 : 2
Complete guide
What is a ratio?
A ratio is a comparison of two quantities that describes how much of one thing there is relative to another. Written A:B (read "A to B"), it expresses the relative sizes of two amounts — not their actual values. A ratio of 3:4 means "for every 3 of one thing, there are 4 of the other." Ratios appear in everyday contexts: mixing paint, scaling recipes, sharing profits, and map scales.
How to simplify a ratio
Simplifying a ratio means dividing both parts by their Greatest Common Divisor (GCD) until neither part can be divided further without producing a non-integer result. The simplified form expresses the same relationship with the smallest possible integers:
GCD(12, 8) = 4
12 ÷ 4 = 3
8 ÷ 4 = 2
12 : 8 = 3 : 2
For decimal ratios, first multiply both parts by a power of 10 to convert to integers, then find the GCD:
Multiply by 10 → 15 : 25
GCD(15, 25) = 5
15 ÷ 5 = 3, 25 ÷ 5 = 5
1.5 : 2.5 = 3 : 5
Equivalent ratios
Two ratios are equivalent if they simplify to the same lowest-terms form. Multiplying or dividing both parts of a ratio by the same non-zero number produces an equivalent ratio:
(each simplifies to 2:3)
Equivalent ratios are the foundation of map scales, recipe scaling, and unit rates. If a recipe for 4 people uses 2 cups of flour : 3 cups of milk, the same ratio scaled up for 12 people (×3) uses 6 cups of flour : 9 cups of milk.
How to solve a proportion (A:B = C:D)
A proportion is a statement that two ratios are equal: A:B = C:D. When three of the four values are known, the fourth can be found using cross-multiplication:
Cross-multiply: A × D = B × C
Solving for each unknown:
A = B × C / D
B = A × D / C
C = A × D / B
D = B × C / A
Example: A car travels 3 km in 4 minutes. At the same speed, how far does it travel in 10 minutes?
Cross-multiply: ? × 4 = 3 × 10
? = 30 ÷ 4 = 7.5 km
How to divide a quantity in a ratio
Dividing a quantity in a ratio A:B means finding what fraction of the total each part represents:
Value per part = Total ÷ (A + B)
Part A = Value per part × A
Part B = Value per part × B
Example: Divide $250 in the ratio 3:2.
Value per part = $250 ÷ 5 = $50
Part A = $50 × 3 = $150
Part B = $50 × 2 = $100
Check: $150 + $100 = $250 ✓
Common ratio examples across contexts
| Context | Ratio | Simplified | Meaning |
|---|---|---|---|
| Aspect ratio | 16:9 | 16:9 | Widescreen display standard |
| Map scale | 1:50000 | 1:50000 | 1 cm = 500 m on the map |
| Recipe scaling | 2:3 | 2:3 | Flour to sugar ratio |
| Profit sharing | 60:40 | 3:2 | Partner A gets 60%, B gets 40% |
| Mortar mix | 1:3 | 1:3 | 1 part cement, 3 parts sand |
| Golden ratio | 1.618:1 | — | Beauty in art and architecture |
| Photo print | 4:6 | 2:3 | Standard 4×6 inch photo |
| Probability | 1:4 | 1:4 | 1 in 5 chance (20%) |
Ratio vs. fraction vs. percentage
Ratios, fractions, and percentages all express relationships between quantities; they are different representations of the same information:
| Ratio | Fraction | Decimal | Percentage |
|---|---|---|---|
| 1:1 | 1/2 | 0.5 | 50% |
| 1:3 | 1/4 | 0.25 | 25% |
| 3:1 | 3/4 | 0.75 | 75% |
| 1:4 | 1/5 | 0.20 | 20% |
| 2:3 | 2/5 | 0.40 | 40% |
| 3:2 | 3/5 | 0.60 | 60% |
The fraction is A/(A+B) — the share of the first part relative to the total.
The GCD algorithm (Euclidean method)
The Greatest Common Divisor is found by the Euclidean algorithm — repeatedly replacing the larger number with the remainder of dividing the larger by the smaller, until the remainder is zero:
48 = 2 × 18 + 12 → GCD(18, 12)
18 = 1 × 12 + 6 → GCD(12, 6)
12 = 2 × 6 + 0 → GCD = 6
48 : 18 ÷ 6 = 8 : 3 (simplified)
Scaling ratios for recipes and models
Ratios are perfect for scaling any recipe or model. Given an original ratio and a desired scale factor, multiply both parts by the factor:
Scaling to 10 servings (factor = 2.5):
flour = 2 × 2.5 = 5 cups
sugar = 3 × 2.5 = 7.5 cups
Use the Scale factor input in Simplify mode to compute scaled values instantly. The calculator shows the simplified form alongside the scaled form so you can choose the most useful representation.
Worked proportion examples
Proportions appear constantly in everyday problem-solving:
- Speed and distance: If a car travels 120 km in 2 hours, how far does it travel in 5 hours? 120:2 = ?:5 → ? = 120 × 5 / 2 = 300 km.
- Currency conversion: If £1 = $1.27, how many dollars is £350? 1:1.27 = 350:? → ? = 1.27 × 350 / 1 = $444.50.
- Scaling diagrams: A blueprint at 1:50 shows a room as 6 cm. Real dimension = 6 × 50 = 300 cm = 3 metres.
- Ingredient substitution: A recipe calls for 2 eggs per 3 cups of flour. For 5 cups, how many eggs? 2:3 = ?:5 → ? = 2 × 5 / 3 = 3.33 eggs.