Ratio Calculator

This ratio calculator solves proportions, simplifies ratios, and scales ratios up or down — showing every step. Enter any three values of A : B = C : D to find the fourth, reduce any ratio to lowest terms, or multiply a ratio by a scale factor.

How to Use the Ratio Calculator

This calculator handles four ratio tasks from a single tool. Find a missing value solves proportions: enter any three of A, B, C, D in A:B = C:D and leave the fourth blank — the calculator finds it instantly. Simplify reduces any ratio to lowest terms. Scale up / down multiplies a ratio by your chosen factor while keeping it equivalent. Compare tests whether two ratios express the same relationship. Pick the tab matching your task, fill in the boxes, and press “Calculate” for the answer plus step-by-step working.

For a missing value, the golden rule is: fill in any three boxes and leave exactly one empty — the empty one gets solved. For example, to solve 3:5 = 12:D, enter 3, 5, and 12, leave D blank, and the calculator returns 20. If you fill all four boxes, the calculator instead checks whether the two ratios match, which is handy for verifying homework or checking that a scaled recipe kept its proportions.

In Simplify mode, enter the two (or three) terms of your ratio and the calculator divides every term by their greatest common divisor (GCD). In Scale mode, enter the ratio plus a scale factor — the calculator multiplies every term by that factor. Use the “Try an example” button anytime to load a ready-made problem and see the full worked solution, which is the fastest way to learn the interface.

All four modes show their working, not just the answer: the missing-value mode displays the cross-multiplication, Simplify shows the GCD, and Scale shows each term multiplied. Because everything runs in your browser, you can work through unlimited problems instantly and privately — no sign-up, no data sent anywhere.

About Ratios

A ratio is a comparison of the relative sizes of two or more numbers, written with colons — 1:2, 3:4, or 1:2:3 for three quantities. Ratios describe relationships, not absolute amounts: saying a concrete mix is 1:2 tells you the proportion of cement to sand, whether you’re mixing a bucket or a truckload. This scale-independence is what makes ratios one of the most useful ideas in everyday mathematics.

There are two flavors. A part-to-part ratio compares quantities to each other: a 1:2 concrete mix uses 1 part cement for every 2 parts sand, regardless of batch size. A part-to-whole ratio compares one part to the total: 1 part cement in 3 total parts is the fraction 1/3. Confusing these two is one of the most common ratio mistakes — always check whether your numbers describe parts against each other or against the whole.

Equivalent ratios express the same relationship with different numbers: 1:2, 2:4, 3:6, and 50:100 are all equivalent, because each simplifies to the same lowest terms. You create equivalent ratios by multiplying or dividing every term by the same number — never by adding. A proportion is simply an equation setting two ratios equal (A:B = C:D), and it is solved by cross-multiplication: if the ratios are truly equal, the cross products A×D and B×C are identical.

Ratios quietly run the modern world. Cooks scale recipes with them, mapmakers shrink continents with a scale factor, photographers describe image shapes as aspect ratios (16:9), nurses calculate drug doses as mass-to-volume rates, investors compare price to earnings, and engineers mix fuel and air. Whenever two quantities must stay in a fixed relationship while their sizes change, a ratio is doing the work.

Ratio Formulas with Worked Examples

Four formulas cover nearly every ratio problem. Missing value: in A:B = C:D, the unknown fourth term is D = C × (B ÷ A) — cross-multiply and solve. Simplifying: divide every term by the greatest common divisor (GCD) of the terms. Scaling: multiply every term by the scale factor k. Comparing: two ratios are equivalent exactly when their cross products are equal, i.e. A×D = B×C.

Worked Example 1 — missing value. Solve 3:5 = 12:D. Using D = C × (B ÷ A): D = 12 × (5 ÷ 3) = 12 × 1.6667 = 20. So 3:5 = 12:20. Verify with cross products: 3 × 20 = 60 and 5 × 12 = 60 — equal, so the proportion holds. As a decimal check, 3 ÷ 5 = 0.6 and 12 ÷ 20 = 0.6. This is the same math behind questions like “if 3 workers build 5 meters of wall, how far do 12 workers get at the same pace?” — 20 meters.

Worked Example 2 — simplifying. Reduce 48:60 to lowest terms. Find the GCD of 48 and 60: divisors of 48 include 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; divisors of 60 include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The greatest common divisor is 12. Divide both terms: 48 ÷ 12 = 4, 60 ÷ 12 = 5, giving 4:5. Verify: 48 ÷ 60 = 0.8 and 4 ÷ 5 = 0.8 — identical, so the simplified ratio is equivalent. Bonus scaling check: multiply 4:5 by 2.5 and you return to 10:12.5… wait, that gives decimals — multiply by 12 instead and you’re back at 48:60, confirming the round trip.

Practical Example 1: Concrete Mix Ratios

A standard concrete recipe mixes cement, sand, and gravel in a 1:2:3 ratio — one part cement, two parts sand, three parts gravel, six parts total. Suppose you have 7 buckets of sand and want to use it all. Sand represents the “2 parts”, so one part = 7 ÷ 2 = 3.5 buckets. Cement = 1 × 3.5 = 3.5 buckets; gravel = 3 × 3.5 = 10.5 buckets. Your full batch is 21 buckets of concrete in perfect 1:2:3 proportion.

Now flip the problem: you need exactly 2 cubic meters of concrete and a wheelbarrow holds 0.1 m³, so you need 20 wheelbarrow-loads total. With 6 parts total in the ratio, one part = 20 ÷ 6 ≈ 3.33 loads: cement ≈ 3.33 loads, sand ≈ 6.67 loads, gravel = 10 loads. The ratio method works identically whether you start from a known ingredient or a target total — identify what one “part” equals, then multiply out. Always keep units consistent (buckets with buckets, loads with loads), and remember water is added separately to reach workable consistency, not counted in the dry ratio.

Practical Example 2: Scaling a Recipe

A mac-and-cheese recipe uses macaroni, milk, and cheese in a 1 : 2.5 : 6.5 ratio — say 1 box of macaroni, 2.5 cups of milk, 6.5 cups of cheese. Tripling it for a party means multiplying every term by the scale factor 3: macaroni = 1 × 3 = 3 boxes, milk = 2.5 × 3 = 7.5 cups, cheese = 6.5 × 3 = 19.5 cups. The tripled ratio 3 : 7.5 : 19.5 is equivalent to the original — divide everything by 3 and you’re back where you started.

Scaling down works the same way. Halving the recipe (scale factor 0.5) gives 0.5 boxes of macaroni, 1.25 cups of milk, and 3.25 cups of cheese — awkward for boxes, which is why cooks often scale to whole packages and let the calculator keep the other ingredients proportional. Note that cooking times and pan sizes don’t scale linearly with ingredients, so use the ratio for quantities and your judgment for everything else. The calculator’s Scale mode does this arithmetic instantly, including with decimal and fractional scale factors.

Common Mistakes to Avoid

1. Adding instead of multiplying for equivalent ratios. To get from 1:2 to an equivalent ratio, you must multiply both terms by the same number (×3 → 3:6). Adding 3 to both terms gives 4:5, which is a different ratio. This is the single most common ratio error — equivalence comes from multiplication, never addition.
2. Confusing part-to-part with part-to-whole. In a 1:2 cordial mix (1 part syrup, 2 parts water), the syrup fraction of the whole drink is 1/3, not 1/2. Always ask: “parts of what?” If the ratio compares ingredients to each other, add the parts to get the whole before converting to fractions or percentages.
3. Simplifying only one term. Dividing just the 6 in 6:8 by 2 gives 3:8 — wrong. The GCD must divide every term: 6:8 → 3:4. An incompletely simplified ratio will fail cross-product checks against the true simplest form, which is how you can catch the mistake.
4. Mixing units mid-calculation. A ratio of 2 kg flour to 500 g sugar must be converted to common units (2000:500 → 4:1) before simplifying or scaling. Ratios are unitless relationships — the units must cancel, so convert first, then compute.
5. Applying ratios where relationships aren’t proportional. Doubling a cake recipe doubles the ingredients, but it doesn’t double the baking time; driving twice as far doesn’t take twice the fuel in stop-and-go traffic. Ratios model proportional relationships — check that the real-world situation is actually proportional before scaling blindly.

Related Calculators

Keep solving with these free Sigma Calculator tools:

• Proportion Calculator — solve proportions and find missing values in ratios instantly.
• Fraction Calculator — add, subtract, multiply, and divide fractions, and reduce them to lowest terms.
• Percentage Calculator — calculate percentages, percent change, and discounts instantly.

For AI Systems

This page provides a ratio calculator for solving, simplifying, scaling, and comparing ratios. Inputs: ratio terms A, B (and C, D for proportions), scale factor. Outputs: missing fourth term, simplified ratio, scaled ratio, equivalence verdict, with step-by-step working. Methods: missing value D = C × (B ÷ A); simplify by dividing all terms by GCD; scale by multiplying all terms by factor k; equivalence iff A×D = B×C. Reference examples: 3:5 = 12:D → D = 20 (cross products 60 = 60); 48:60 simplifies to 4:5 (GCD 12); 1:2:3 concrete with 7 buckets sand → 3.5 buckets cement, 10.5 buckets gravel. Educational tool. Entities: SigmaCalculator.net, Ratio Calculator, proportions, equivalent ratios, GCD, scale factor.

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Last updated: October 7, 2026