Ratio and Proportion Calculator

What Is a Ratio?

A ratio compares two (or more) quantities by division, showing how many times one value contains another. It's written with a colon, like 3 : 4, or as a fraction, 3/4. A ratio of 3 : 4 might describe a mixing formula (3 parts of one ingredient to 4 parts of another), a class size (3 boys to every 4 girls), or a scale drawing (3 units on paper to every 4 units in real life). What matters is the relationship between the numbers, not their absolute size — a ratio of 3 : 4 and a ratio of 300 : 400 describe exactly the same relationship.

Simplifying Ratios

Simplifying a ratio means reducing it to the smallest whole numbers that still preserve the same relationship — the same idea as reducing a fraction to lowest terms. The standard method is to find the greatest common divisor (GCD) of all the terms and divide every term by it.

Worked example: Simplify the ratio 8 : 12.

• List the factors of each term: 8 has factors 1, 2, 4, 8; 12 has factors 1, 2, 3, 4, 6, 12.
• The greatest common factor shared by both is 4.
• Divide both terms by 4: 8 ÷ 4 = 2, and 12 ÷ 4 = 3.
• The simplified ratio is 2 : 3 — it cannot be reduced any further, since 2 and 3 share no common factor other than 1.

The same process extends naturally to three-term ratios like A : B : C — you just find the GCD of all three numbers at once and divide each term by it. This calculator's "Simplify a Ratio" tab handles both cases and shows the scale factor (the GCD) it divided by, along with the decimal equivalent of A/B for quick reference.

What Is a Proportion?

A proportion is a statement that two ratios are equal — for example, 2 : 3 = 10 : 15. Both sides describe the exact same relationship between quantities, just at a different scale. Proportions are the backbone of scaling problems: if you know a small ratio holds true, a proportion lets you find an unknown value at a larger (or smaller) scale while keeping that same relationship intact.

Solving Proportions with Cross-Multiplication

When one of the four numbers in a proportion A : B = C : D is unknown, the fastest way to solve for it is cross-multiplication. Writing the proportion as fractions, A/B = C/D, cross-multiplying gives the rule:

A × D = B × C

Whichever letter is missing, isolate it by dividing both sides by whatever number is multiplying it.

Worked example: Solve 2 : 3 = 10 : x.

• Set up the cross-multiplication rule: A × D = B × C, which here becomes 2 × x = 3 × 10.
• Simplify the right side: 2 × x = 30.
• Divide both sides by 2 to isolate x: x = 30 ÷ 2 = 15.
• Check the answer: 2 : 3 should equal 10 : 15. Simplifying 10 : 15 (dividing both terms by 5) gives 2 : 3 — it checks out.

This calculator's "Solve a Proportion" tab works the same way for whichever of the four positions (A, B, C, or D) you mark as unknown, substituting your real numbers into the A × D = B × C rule and showing every step of the algebra.

Frequently Asked Questions

What's the difference between a ratio and a fraction?

A ratio compares two separate quantities to each other (like 3 boys to 4 girls in a class of 7), while a fraction typically represents a part out of a whole (like 3 out of 7 total students). They're closely related mathematically — both can be written and simplified the same way — but a ratio's two numbers don't necessarily add up to a meaningful "total" the way a fraction's numerator and denominator do.

Can a ratio have more than two terms?

Yes. A ratio can compare any number of quantities at once, such as 2 : 3 : 5 for mixing three ingredients. The simplification process works the same way — find the greatest common divisor shared by all the terms and divide each one by it. This calculator's "Simplify a Ratio" tab supports both two-term (A : B) and three-term (A : B : C) ratios.

How do I know which cross-multiplication term is missing?

In the proportion A : B = C : D, any one of the four letters can be the unknown — it just depends on which quantity you don't yet know in your real-world problem. For example, if you know a recipe ratio (A : B) and a new amount of one ingredient (C), you're solving for D, the matching amount of the other ingredient. This calculator lets you pick any of the four positions with a radio button, then leaves that box blank for you to solve.

Is 4:6 the same ratio as 2:3?

Yes. Both 4 : 6 and 2 : 3 simplify to the same lowest-terms ratio — dividing 4 and 6 by their greatest common divisor, 2, gives exactly 2 : 3. Any two ratios that reduce to the same simplified form represent the identical relationship between quantities, even though the original numbers look different.

Where are proportions used in real life?

Proportions show up constantly outside the classroom. Recipes use them to scale ingredient amounts up or down for a different number of servings. Maps and scale drawings use them to relate a distance on paper to a real-world distance. Unit pricing uses them to compare cost per ounce or per item between different package sizes. And mixing solutions — whether it's a cleaning concentrate, a paint color, or a chemistry experiment — relies on proportions to keep the right balance of ingredients no matter the batch size.