Ratio Calculator
Three ratio tools in one: simplify a ratio, solve for a missing value in a proportion, or convert a ratio to percentages. All results show step-by-step work.
Simplify a Ratio
A : B = 12 : 8
GCD(12, 8) = 4
Simplified = (12 ÷ 4) : (8 ÷ 4)
= 3 : 2Solve Equivalent Ratio (A : B = C : D)
Enter three values — choose which one is missing.
A : B = C : D 3 : 4 = 9 : D Cross-multiply: A × D = B × C 3 × D = 4 × 9 3 × D = 36 D = 36 / 3 D = 12
Ratio to Percentage
Ratio A : B = 3 : 2 Total parts = 3 + 2 = 5 A as a percentage: A% = A / Total × 100 A% = 3 / 5 × 100 A% = 60% B as a percentage: B% = B / Total × 100 B% = 2 / 5 × 100 B% = 40% Check: 60% + 40% = 100%
What Is a Ratio?
A ratio expresses the relationship between two quantities of the same kind. It answers the question "how much of one thing is there compared to another?" Ratios appear everywhere: recipe proportions (2 cups flour : 1 cup sugar), maps (1:50,000 means 1 cm represents 500 m), financial analysis (price-to-earnings ratio), and mixing of solutions in chemistry and engineering.
Ratios can be part-to-part (comparing two parts directly, like 3 boys to 4 girls) or part-to-whole (comparing one part to the total, like 3 boys out of 7 students). The part-to-whole form is equivalent to a fraction.
How to Simplify a Ratio
Simplifying a ratio is identical to simplifying a fraction: find the GCD of both terms and divide each by it. The result expresses the same proportion with the smallest possible whole numbers.
Formula:
Simplified A : B = (A ÷ GCD) : (B ÷ GCD)
The Euclidean algorithm efficiently finds the GCD: repeatedly divide the larger number by the smaller and replace the larger with the remainder, until the remainder is zero. The last non-zero remainder is the GCD.
Example: Simplify 48:36
GCD(48, 36): 48 = 1 × 36 + 12; 36 = 3 × 12 + 0. GCD = 12.
48 ÷ 12 = 4; 36 ÷ 12 = 3
Simplified: 4 : 3
Uses: simplifying a mix ratio for paint (4 parts base : 1 part hardener), reducing a map scale, expressing a survey result in simplest form.
Solving Proportions
A proportion is an equation stating that two ratios are equal: A:B = C:D, or equivalently A/B = C/D. If one of the four values is unknown, you can solve for it using cross-multiplication.
Formula:
A / B = C / D ⇒ A × D = B × C
This works because equal fractions have equal cross-products. Once you have three of the four values, you can always find the fourth.
Example: A car travels 150 miles on 5 gallons. How many miles on 8 gallons?
Set up: 150/5 = x/8
Cross-multiply: 150 × 8 = 5 × x
1200 = 5x
x = 1200 / 5 = 240 miles
Proportions are used in scaling recipes (doubling a recipe for 4 into one for 10), unit conversions (miles to kilometers), map reading, and currency exchange.
Converting Ratios to Percentages
A part-to-part ratio A:B can be converted to percentages by treating the total (A+B) as 100%. Each part becomes a fraction of the whole.
Formula:
A% = A / (A + B) × 100 B% = B / (A + B) × 100
Example: A solution is mixed in a 3:7 ratio of alcohol to water.
Total parts = 3 + 7 = 10
Alcohol = (3/10) × 100 = 30%
Water = (7/10) × 100 = 70%
Check: 30% + 70% = 100%. The percentages always sum to 100 in a two-part ratio.
This conversion is useful in business (market share split), nutrition (macronutrient breakdown), project management (resource allocation), and statistics (category proportions).
Common Ratio Examples and Applications
| Context | Ratio | Meaning |
|---|---|---|
| Cement mix (basic) | 1:2:3 | 1 part cement, 2 sand, 3 aggregate |
| Golden ratio | 1:1.618 | Appears in nature and art |
| Screen aspect ratio (HD) | 16:9 | Width to height |
| Map scale | 1:25,000 | 1 cm = 250 m in real life |
| PE ratio (stock) | 25:1 | Price is 25 times earnings |
| Debt-to-equity | 0.4:1 | $0.40 debt per $1 equity |
Frequently Asked Questions
What is a ratio and how is it written?
A ratio compares two quantities, showing how many times one value contains or is contained in another. It can be written three ways: using a colon (3:4), as a fraction (3/4), or in words ("3 to 4"). All three representations mean the same thing. Unlike fractions, ratios do not require a whole — the two quantities can each stand on their own. For example, "the ratio of boys to girls is 3:4" means for every 3 boys there are 4 girls.
How do you simplify a ratio?
To simplify a ratio A:B, find the greatest common divisor (GCD) of A and B, then divide both by it. For example, 12:8. GCD(12,8) = 4. Simplified: (12÷4):(8÷4) = 3:2. A simplified ratio has the same proportional relationship as the original — just expressed with the smallest possible whole numbers.
How do you solve a proportion (find a missing value in an equivalent ratio)?
A proportion states that two ratios are equal: A/B = C/D. To find the missing value, use cross-multiplication. If D is unknown: A × D = B × C, so D = (B × C) / A. Example: 3/4 = 9/D → D = (4 × 9) / 3 = 36 / 3 = 12. The proportion 3:4 = 9:12 is correct because both simplify to 3:4.
How do you convert a ratio to a percentage?
To convert ratio A:B to percentages, find the total parts (A + B), then divide each part by the total and multiply by 100. For ratio 3:2: Total = 5. A% = (3/5) × 100 = 60%. B% = (2/5) × 100 = 40%. The percentages always add up to 100%. This is used in situations like "what share of the mixture is each ingredient?"
What is the difference between a ratio and a fraction?
A fraction represents a part of a whole — the denominator is the total and the numerator is a part of it. A ratio compares two separate quantities that do not have to add up to a whole. However, when a ratio is used in a part-to-whole sense (like 3 red balls out of 5 total), it can be expressed as a fraction (3/5). The ratio 3:2 (part-to-part) becomes the fraction 3/5 (part-to-whole) only when you define the whole as 3+2=5.
Related Calculators
- Fraction Calculator — add, subtract, multiply, divide fractions with steps
- Percentage Calculator — three percentage problem types