Ratio Simplifier and Proportion Solver
Free tool that reduces ratios like 12:18 to 2:3 and solves proportions like 3:4 = x:12 for x, with step-by-step working. Handles three-term ratios such as 4:6:10 and fractional answers.
How to use
- Choose a mode: “Simplify a ratio” or “Solve a proportion”.
- Enter a ratio like 12:18, or a proportion like 3:4 = x:12.
- Press “Calculate” to see the answer and the steps.
Examples
- 12 : 18 → 2 : 3
- 4 : 6 : 10 → 2 : 3 : 5
- 3 : 4 = x : 12 → x = 9
When to use it (grade level)
Ratios are taught in earnest in 6th grade and lead into proportion, percentages, and similar figures in middle school. This tool suits checking home study, reviewing worksheets, and confirming exam-style calculation practice.
A ratio expresses the relationship between two or more quantities, written like “3:4”. This tool has two modes. One simplifies a ratio, turning something like 12:18 into its simplest whole-number form, 2:3. The other solves a proportion, finding x in an equation like 3:4 = x:12 (here x = 9).
How to work it out yourself
To simplify a ratio, divide every term by the same number: the greatest common divisor of all the terms. For 12:18, the greatest common divisor of 12 and 18 is 6, so 12÷6 : 18÷6 = 2:3. The same works for three or more terms: 4:6:10 divides by the shared factor 2 to give 2:3:5.
To solve a proportion a:b = c:d, use the rule “the product of the means equals the product of the extremes”: the outer pair (a and d) multiplied together equals the inner pair (b and c) multiplied together. For 3:4 = x:12, the means 4×x equal the extremes 3×12, so 4x = 36, and dividing both sides by 4 gives x = 9.
How ratios relate to division and fractions
A ratio is closely tied to division and fractions. The ratio “3:4” means the first number divided by the second: 3÷4 = 3/4. This 3/4 is called the value of the ratio. Simplifying a ratio does not change its value: 6:8 simplifies to 3:4, and 6÷8 = 3/4, so the value stays the same.
That means a proportion can be read as an equation of fractions. 3:4 = x:12 is the same as 3/4 = x/12 — both describe the same relationship of size. Solving gives x = 9, and reducing 9/12 gives 3/4, confirming it has the same ratio value as 3:4. When the answer does not come out whole, as in 2:3 = 5:x where x = 15/2, keeping it as a fraction is the exact result.
Common mistakes
Three slip-ups are common. First, simplifying only part of the ratio: dividing just one side, as in writing 4:6 as 2:4, breaks the ratio. Divide both by the same number (here 2) to get 2:3. Second, stopping at a small common factor and leaving a form that can still be reduced further, rather than using the greatest common divisor.
Third, mixing up the pairs in “means × means = extremes × extremes”. In a:b = c:d, the extremes are the outer terms a and d, and the means are the middle terms b and c. Getting the positions wrong throws off the answer, so rewriting as a fraction equation like 3/4 = x/12 before solving helps avoid errors.
Related topics and tools
Ratios go hand in hand with greatest common divisors and fractions. Simplifying a ratio uses exactly the same mechanics as reducing a fraction, so pairing this with the GCD and LCM calculator and the fraction calculator deepens your understanding. For practice expressing a ratio value like 3/4 as a fraction, the fraction calculator is handy. Decide yourself what to divide by and which terms are the means and extremes, then use this tool to check the steps.
FAQ
Can it simplify ratios with three or more terms?
Yes. A ratio like 4:6:10 is divided by the greatest common divisor of all the terms to reach the simplest whole-number form, 2:3:5. Up to 10 terms are supported.
Can x be in any position of the proportion?
Yes. Whether you write x:4 = 3:12 or 3:4 = x:12, x can be in any of the four positions. For 3:4 = x:12, the answer is x = 9.
What happens when the answer doesn’t divide evenly?
It is shown exactly as a fraction. For example, 2:3 = 5:x gives x = 15/2 as a reduced fraction.