Math

Ratio Calculator

Simplify a ratio down to its lowest terms, or solve for the missing number in a proportion — either way, see the full working, not just the final answer.

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What is Ratio Calculator?

A ratio compares two quantities, showing how many times one contains the other. Ratios can always be simplified to their lowest terms by dividing both numbers by their greatest common divisor (GCD) — just like simplifying a fraction. A proportion is a statement that two ratios are equal, written as a:b = c:x. Proportions are commonly used to scale recipes, convert measurements, adjust map distances, or solve for an unknown quantity when you know it's proportional to a known one. This calculator both simplifies ratios and solves proportions for a missing value using cross-multiplication.

When to Use This Calculator

  • Simplify any ratio like 24:36 to its lowest terms before using it in a recipe, mix, or design
  • Check whether two ratios are equal without calculating their decimal values
  • Find a missing value in real-life proportions such as recipes, map scales, or mix ratios
  • Present a two-part comparison as a percentage split when reporting data
  • Solve school homework on ratios, rates, and proportional reasoning
  • Verify per-unit price and value comparisons when shopping for the best deal

Steps:

  1. Choose Simplify Ratio mode to reduce a:b to its lowest terms.
  2. Or choose Solve Proportion mode to find a missing value in a:b = c:x.
  3. Enter the known values.
  4. Review the full step-by-step calculation.

Formula

Simplify: a:b = (a÷GCD(a,b)) : (b÷GCD(a,b)) Decimal: a/b Proportion: a/b = c/x, solved as x = (b × c) / a

Use Cases

  • Simplifying recipe ingredient ratios when scaling up or down
  • Solving map-scale and blueprint-scale proportions
  • Business and finance ratios, like price-to-earnings or debt-to-income
  • Math homework involving ratios, rates, and proportional reasoning

Key Benefits

  • Simplifies any ratio to its lowest terms using the greatest common divisor
  • Solves proportions for a missing value using cross-multiplication
  • Shows the ratio as a decimal and as a percentage split
  • Full step-by-step solution for every calculation, not just the final answer

Pro Tips

  • To scale a ratio up, multiply both numbers by the same value; to simplify it, divide both by their GCD
  • If a ratio can't be simplified further, its two numbers are already coprime (their GCD is 1)
  • When solving a proportion, always double-check that the units are consistent on both sides before cross-multiplying
  • A ratio of 1:1 means the two quantities are exactly equal

Common Mistakes to Avoid

  • Simplifying only one side of a ratio instead of dividing both numbers by the same GCD
  • Setting up a proportion with mismatched units on each side (e.g., mixing minutes with hours)
  • Cross-multiplying incorrectly by multiplying the wrong pair of terms
  • Confusing a ratio (a comparison, a:b) with a fraction of a whole (a part-to-whole relationship, like a/(a+b))

Key Terms Explained

Ratio: A comparison of two quantities showing their relative sizes, written as a:b
Proportion: A statement that two ratios are equal, written as a:b = c:d
Greatest Common Divisor (GCD): The largest number that divides both terms of a ratio without a remainder
Cross-Multiplication: A technique for solving proportions by multiplying diagonally across the equals sign

Related Concepts

  • Ratios written as fractions can be added, subtracted, and compared with the Fraction Calculator.
  • Turn any ratio or part of a whole into its percentage form with the Percentage Calculator.
  • Solve more general equations involving unknown quantities with the Equation Solver.
  • Finding the greatest common divisor, the key step in simplifying ratios, is covered by the GCD/LCM Calculator.
  • Convert the units used in a proportion, such as kilometers to miles, with the Unit Converter.

Example

Simplifying 12:18 — the GCD of 12 and 18 is 6, so the simplified ratio is 2:3. Solving the proportion 4:6 = 10:x — cross-multiplying gives 4x = 60, so x = 15.

Interpreting Your Results

The simplified ratio a:b shows the two quantities in their lowest whole-number form. The decimal equivalent a/b is the scale factor between them, so the ratio 3:2 has a decimal of 1.5, meaning the first quantity is one and a half times the second. The percentage split a/(a+b) and b/(a+b) shows each term's share of the whole, and the two shares always total 100%. When you solve the proportion a/b = c/x, the answer x = (b × c)/a is the value that keeps both ratios equal.

Frequently Asked Questions

How do I simplify a ratio?
Find the greatest common divisor (GCD) of both numbers, then divide each number by that GCD. For example, to simplify 12:18, the GCD is 6, so dividing both by 6 gives the simplified ratio 2:3.
How do I solve a proportion for an unknown value?
Use cross-multiplication. For the proportion a/b = c/x, multiply a by x and b by c to get a × x = b × c, then divide both sides by a to solve for x: x = (b × c) / a.
What is the difference between a ratio and a fraction?
A ratio compares two separate quantities (a:b), while a fraction typically represents a part of a whole (a/(a+b)). However, a ratio a:b can always be converted into the fraction a/b to express it as a single number.
Can a ratio have more than two numbers?
Yes — ratios can compare three or more quantities at once, such as 2:3:5 for mixing paint or ingredients. This calculator focuses on two-term ratios (a:b), which are the most common case, but the same simplification principle (dividing by the GCD of all terms) applies to ratios with more parts.
How do I check whether two ratios are equivalent?
Two ratios are equivalent when their cross-products are equal, or when they simplify to the same lowest terms. For example, 2:3 and 4:6 are equivalent because 2 × 6 = 12 and 3 × 4 = 12. In the same way, 3:4 and 6:8 give 3 × 8 = 24 and 4 × 6 = 24, so they are also equivalent. If the cross-products differ, the ratios are not equal.
How do I divide an amount in a given ratio?
Add the ratio parts to find the total number of shares, then divide the amount by that total to find the value of one share. To split $120 in the ratio 3:2, add 3 + 2 = 5 shares, so each share is 120 ÷ 5 = 24. The first part receives 3 × 24 = 72 and the second receives 2 × 24 = 48, giving $72:$48.
How do I express a ratio as a percentage?
First write each ratio term as a fraction of the whole, then multiply by 100. For the ratio 3:2, the whole is 3 + 2 = 5 parts. The first term is 3/5 = 0.6 = 60% and the second term is 2/5 = 0.4 = 40%. The two percentages always add up to 100%, so you can find the second share by subtracting the first from 100.
What is a unit rate, and how is it different from a ratio?
A unit rate compares a quantity to one unit of another quantity, such as speed in miles per hour, and is found by dividing the two numbers. The ratio 240 km : 3 h simplifies to the unit rate 80 km per hour. Ratios compare two quantities without reducing to a single unit, while a unit rate always has a denominator of 1.
Can a ratio contain decimals or fractions?
Yes. Multiply every term by the same number to clear the decimals or fractions first, then simplify. For the ratio 0.5:1.5, multiplying both terms by 2 gives 1:3, which is fully simplified. For (1/2):(1/3), multiply both terms by 6 to get 3:2. Any ratio with fractional or decimal terms can be written with whole numbers this way.
Why do I divide by the GCD when simplifying a ratio?
The GCD is the largest number that divides both terms, so dividing by it removes every common factor at once and leaves the ratio in lowest terms. For 12:18, the GCD is 6, giving 2:3. Dividing by a smaller common factor, such as 2, would leave 6:9, which can still be simplified. Only the GCD produces a ratio whose two terms are coprime.
What does the reciprocal of a ratio represent?
Swapping the two terms gives the reciprocal ratio. The reciprocal of 2:3 is 3:2, and it describes the same two quantities read in the opposite order. Reciprocals are useful when a relationship is reversed, such as a gear ratio where a 3:1 driver-to-driven ratio becomes 1:3 when looked at from the other side. Multiplying a ratio by its reciprocal always gives 1.

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