Ratio Calculator with Simplification and Steps
Simplify two-part or multi-part ratios made from whole numbers, decimals, fractions or mixed numbers. Compare two known ratios, divide an optional total, and review percentages, unit rates and exact steps.
Last Updated: July 26, 2026
Online Ratio Calculator
Use a colon, comma or the word “to” between values. The optional fields divide a total and test whether another two-part ratio is equivalent.
Your simplified ratio and calculation details will appear here.
Part-to-whole breakdown
| Part | Simplified value | Fraction of whole | Percentage |
|---|---|---|---|
| Calculate a ratio to see each part. | |||
Equivalent ratios
| Scale | Equivalent ratio | Why it is equivalent |
|---|---|---|
| Equivalent ratios will appear here. | ||
Divide the total in this ratio
| Part | Calculation | Allocated amount |
|---|
Calculation steps
- Enter a ratio and select Calculate Ratio.
Recent calculations
- Your last six results will appear here.
How to Use This Ratio Calculator
- Enter two to eight values in the Ratio values field. Separate them with a colon, comma or the word “to.”
- Use whole numbers, decimals, simple fractions or positive mixed numbers. Examples include 12:18, 2.5:3.75 and 1/2:3/4.
- Optionally enter a total if you want to divide an amount according to the ratio.
- Optionally enter a second two-part ratio if you want to test equivalence or compare the first-to-second quotients.
- Select Calculate Ratio. Review the simplest integer form, part percentages, equivalent ratios, allocations and calculation steps.
Keep the terms in their intended order. A ratio of 2:5 does not mean the same thing as 5:2. Avoid thousands separators inside a term because a comma separates ratio values. Enter 1500 rather than 1,500.
What Is a Ratio?
A ratio compares the relative sizes of two or more quantities. The ratio 3:2 says there are three units of the first quantity for every two units of the second. It describes a relationship, not necessarily the actual number of objects. The quantities 3 and 2, 6 and 4, or 30 and 20 all have the same 3:2 relationship.
A two-part ratio may be written as 3:2, “3 to 2,” or the quotient 3/2 when the context is clear. A multi-part ratio such as 2:3:5 compares three quantities at once. In all forms, the order and labels matter. If a class has 12 students in Group A and 18 in Group B, A:B is 12:18, while B:A is 18:12.
Ratio Calculator Formulas
| Task | Formula | Purpose |
|---|---|---|
| Simplify whole-number ratio | a:b = (a ÷ GCF):(b ÷ GCF) | Reduce all terms by their greatest common factor. |
| Clear fraction denominators | term × LCM of denominators | Turn fractional terms into whole-number terms before GCF reduction. |
| Part percentage | term ÷ sum of terms × 100 | Express each ratio term as a share of the whole. |
| Divide a total | total × term ÷ sum of terms | Allocate a known total in the stated ratio. |
| Compare a:b and c:d | compare a × d with b × c | Test exact equivalence without rounding decimals. |
How to Simplify a Ratio
To simplify a whole-number ratio, find the greatest common factor shared by every term. Divide every term by that same factor. The resulting terms must stay in the original order.
Worked example: simplify 24:36
24 ÷ 12 = 2
36 ÷ 12 = 3
Simplified ratio = 2:3
The ratio 2:3 is in simplest integer form because 2 and 3 share no whole-number factor greater than 1. Multiplying both terms by the same positive value creates an equivalent ratio. Therefore, 2:3, 4:6, 10:15 and 24:36 all describe the same relative relationship.
How the Calculator Handles Decimals and Fractions
Decimals and fractions need one extra step before GCF reduction. The calculator converts every entered term into an exact fraction. It then finds the least common multiple of the denominators and multiplies every term by it. This clears the denominators without changing the ratio.
Decimal example: simplify 2.5:3.75
The exact fractional forms are 5/2 and 15/4. The least common multiple of 2 and 4 is 4. Multiplying both terms by 4 gives 10:15. Dividing both terms by their GCF of 5 gives 2:3.
Fraction example: simplify 1/2:3/4
The least common multiple of the denominators 2 and 4 is 4. Multiply each term by 4: 1/2 × 4 = 2 and 3/4 × 4 = 3. The exact simplified ratio is 2:3. This approach avoids estimating the fractions with rounded decimal values.
Mixed numbers are supported too. For example, 1 1/2:2 1/4 becomes 3/2:9/4. Clearing denominators gives 6:9, which simplifies to 2:3.
How to Divide a Total in a Ratio
First add the simplified ratio terms to find the total number of shares. Divide the total by that share count, then multiply by each term. This method works for money, ingredients, time, inventory or any other total whose units are consistent.
Worked example: divide 84 in the ratio 3:4
Value of one part = 84 ÷ 7 = 12
First allocation = 12 × 3 = 36
Second allocation = 12 × 4 = 48
The allocations are 36 and 48. Their sum is 84, and 36:48 simplifies back to 3:4. When allocations produce repeating decimals, the calculator shows a rounded decimal plus the exact fractional amount. Real cash payments may need a separate rounding rule so the final rounded amounts still add to the original total.
How to Compare Two Ratios
Two ratios are equivalent when their corresponding quotients are equal. For a:b and c:d, compare the cross-products a × d and b × c. Equal cross-products prove equivalence without depending on rounded decimal displays.
Worked example: compare 4:6 and 10:15
6 × 10 = 60
The cross-products are equal, so 4:6 and 10:15 are equivalent.
If the cross-products differ, the calculator compares the value of first term ÷ second term for each ratio. A larger quotient means more of the first quantity for each unit of the second. This comparison requires a nonzero second term. The calculator compares fully known ratios only. Use a dedicated proportion solver when one term is unknown.
Part-to-Part and Part-to-Whole Ratios
A part-to-part ratio compares one category with another. If a tray contains 3 red counters and 2 blue counters, red:blue is 3:2. A part-to-whole ratio compares one category with the complete set. There are 5 counters in total, so red:all is 3:5 and blue:all is 2:5.
The result table converts each entered term into a fraction and percentage of the whole. For 3:2, the first part is 3/5 or 60%, and the second part is 2/5 or 40%. These percentages describe shares of the total. They are different from the first-to-second quotient, which is 3/2 or 1.5.
Ratio, Rate, Fraction and Proportion
| Concept | What it describes | Example |
|---|---|---|
| Ratio | Relative size of two or more quantities. | 3 cups flour to 2 cups milk = 3:2. |
| Rate | A ratio comparing quantities with different units. | 150 kilometres in 3 hours = 50 km/h. |
| Fraction | A number representing division or part of a whole. | 3/5 of the total is in the first part. |
| Proportion | An equation stating that two ratios are equal. | 2/3 = 10/15. |
| Percentage | A ratio expressed per 100. | 3 of 5 total parts = 60%. |
Keeping these ideas separate prevents a common error. In 3:2, the first-to-second quotient is 150%, but the first part is only 60% of the combined total. The denominator changes from the second part to the sum of both parts.
Unit Rates and Equivalent Ratios
A unit rate rewrites a two-part ratio so one side equals 1. For 6:4, the first-per-second unit rate is 6 ÷ 4 = 1.5, written as 1.5:1. The reverse unit rate is 4 ÷ 6 = 0.666667, or about 0.666667:1. Unit rates help compare prices, speed, productivity and mixture intensity when the original totals differ.
Equivalent ratios preserve the same quotient. Multiply or divide every term by the same nonzero value. Multiplying only one term changes the relationship. The calculator generates ×2 through ×5 versions from the simplest integer ratio, which makes the scaling rule easy to verify.
Working with Three-Part and Multi-Part Ratios
A multi-part ratio compares more than two quantities. Suppose a paint mixture uses red, yellow and white in the ratio 2:3:5. There are 10 total parts. Red is 2/10 or 20%, yellow is 3/10 or 30%, and white is 5/10 or 50%.
To divide 40 litres in that ratio, multiply 40 by each term over 10. The allocations are 8, 12 and 20 litres. The same method extends to the calculator’s limit of eight terms. Ratio comparison is limited to two terms because a single quotient and cross-product direction does not fully describe a multi-part relationship.
Practical Uses for Ratio Calculations
- Recipes: scale ingredient relationships while preserving taste and texture.
- Maps and models: connect drawing distance with real distance using a consistent scale.
- Mixtures: compare components in paint, concrete, cleaning solutions or laboratory preparations.
- Budgets: divide a planned total across categories according to a chosen allocation.
- Business: compare staffing, output, conversions or costs using clearly labelled quantities.
- Education: check simplified forms, equivalent ratios, percentages and multi-step working.
A valid mathematical result still needs correct context. Use quantities measured on the same basis. Converting 2 metres and 50 centimetres directly to 2:50 is wrong because the units differ. Convert first: 2 metres is 200 centimetres, so the ratio is 200:50 = 4:1.
Zero Values, Negative Values and Units
A zero term is permitted when it describes a real absence. The ratio 0:5 simplifies to 0:1. However, a unit rate that divides by zero is undefined, and a ratio containing only zeros has no useful scale, so the calculator rejects it.
This tool rejects negative ratio terms. In most part comparison and allocation contexts, quantities are nonnegative. Signed changes, vectors and financial returns require context-specific treatment and should not be reduced as ordinary sharing ratios.
Units cancel only when both quantities use compatible units. When units differ, the relationship is a rate, such as kilometres per hour or cost per kilogram. The quotient remains useful, but the units must stay attached to the result.
Common Ratio Mistakes
- Reversing the order. Write and label the quantities before entering the terms.
- Dividing only one term. Equivalent ratios require the same multiplier or divisor for every term.
- Mixing units. Convert measurements to a common unit before simplifying a like-for-like ratio.
- Using the second term as the whole. For part percentages, divide by the sum of all terms.
- Rounding too early. Keep fractions exact until the final display or required payment step.
- Forgetting to check allocations. Rounded shares should add back to the stated total.
- Treating a ratio as proof of cause. A numerical relationship does not explain why the quantities differ.
Input Rules and Result Precision
The calculator accepts 2 to 8 ratio terms. Each term may be a nonnegative whole number, decimal, simple fraction, mixed number or scientific-notation value. A term may contain up to 50 digits, and the absolute exponent is limited to 100. Fraction denominators must be nonzero.
Core simplification and equivalence checks use integer-based exact arithmetic. Decimal displays use up to 12 places for rates and percentages and up to 10 places for allocations. A displayed decimal may therefore be rounded even when the internal fraction is exact. The exact fraction appears with an allocation when the decimal does not terminate within the display limit.
Related Calculators
Continue with a focused tool when your problem involves arithmetic, recipe quantities, decimal conversion or a specialized market ratio.
References and Review Basis
This page follows standard arithmetic definitions for ratios, unit rates, percentages and proportion checks. For further study, see OpenStax Prealgebra 2e: Ratios and Rate and OpenStax Prealgebra 2e: Proportions. Examples on this page are educational and use hypothetical quantities.
Frequently Asked Questions
How do I simplify a ratio?
Find the greatest common factor shared by all whole-number terms, then divide every term by it. For 12:18, the GCF is 6, so the simplest ratio is 2:3.
Can this calculator simplify a ratio with decimals?
Yes. It converts each decimal to an exact fraction, clears the denominators and reduces the resulting whole-number ratio. For example, 2.5:3.75 simplifies to 2:3.
Can I enter fractions or mixed numbers in a ratio?
Yes. Use entries such as 1/2:3/4 or 1 1/2:2 1/4. Fraction denominators must be nonzero, and every ratio term must be nonnegative.
How do I divide a total in a ratio?
Add the ratio terms, divide the total by that sum, and multiply by each term. Dividing 84 in the ratio 3:4 gives 84 × 3/7 = 36 and 84 × 4/7 = 48.
How do I know whether two ratios are equivalent?
For a:b and c:d, compare a × d with b × c. The ratios are equivalent when the cross-products are equal and the compared second terms are nonzero.
What is the difference between a ratio and a proportion?
A ratio compares quantities, such as 2:3. A proportion is an equation stating that two ratios are equal, such as 2/3 = 10/15.
What is a unit rate?
A unit rate rewrites a two-part ratio per one unit of the second quantity. The ratio 6:4 has a first-per-second unit rate of 6 ÷ 4 = 1.5.
How do I convert a ratio to percentages?
Add all ratio terms, divide each term by the sum, and multiply by 100. In 3:2, the parts are 3/5 = 60% and 2/5 = 40%.
Can a ratio contain more than two numbers?
Yes. A ratio such as 2:3:5 compares three quantities. This calculator accepts between two and eight terms and shows every term as a share of the whole.
Can a ratio include zero?
Yes, when zero represents an absent quantity. The ratio 0:5 simplifies to 0:1. A ratio made entirely of zeros is invalid, and division by a zero term is undefined.
Do ratio terms need the same units?
Use the same units before simplifying a like-for-like ratio. If the units differ, the result is a rate and the units must remain attached, such as kilometres per hour.
Are my ratio values stored or sent anywhere?
No. The calculation and recent history stay in the current browser page and disappear when it reloads. Standard website analytics may record a page visit, but not the values entered into this calculator.