Free Ratio Calculator — Simplify · Equivalent · Scale

What This Calculator Does

A ratio is a way of comparing two quantities — how much of one thing there is relative to another. Written as A:B, it tells you that for every A units of the first thing, there are B units of the second. Ratios are everywhere: cooking recipes (2 cups flour to 3 cups water), business partnerships (a 60:40 equity split), map scales (1:50,000), aspect ratios (16:9 widescreen), gear teeth, mixing ratios for paint, and inheritance shares. Whenever two related quantities need to stay in proportion as either of them changes, you are doing ratio math.

This calculator covers the three operations that make up almost all real-world ratio work. It can simplify any ratio to its smallest whole-number form (12:18 → 2:3), find an equivalent ratio when one side is scaled up or down (2:3 with target 10 → 10:15), and split a total amount in a given ratio (split $300 in 2:3 → $120 and $180). Behind the scenes it uses the Euclidean algorithm to compute the greatest common divisor — the same 2,300-year-old procedure Euclid wrote down in his Elements, still the fastest way to reduce a ratio that’s ever been found.

The Three Modes Explained

Picking the right mode is half the work. Each one solves a different real-world question, and using the wrong one is the most common ratio mistake we see.

Simplify mode — find the smallest whole-number form

You give the calculator two numbers, A and B, and it returns the same ratio in its lowest terms. 12:18 becomes 2:3. 100:25 becomes 4:1. Use this when you have raw measurements and want a clean, canonical form for documentation, communication, or comparison. Two ratios are equal if and only if they reduce to the same simplified form, so simplification is also the easiest way to check whether two ratios are actually the same.

Equivalent mode — find Y when X is given

You know a ratio A:B, and you have a new value for the A-side called target. The calculator finds the matching value on the B-side that keeps the proportion intact. This is recipe scaling: a recipe says 2 cups flour to 3 cups water, you want to use 10 cups of flour, how much water? The math is target × B / A — in this case 10 × 3 / 2 = 15 cups. The same logic powers map distances, currency conversion, scaled drawings, and any cross-multiplication problem.

Scale mode — split a total in a ratio

You give the calculator a ratio A:B and a total, and it splits the total into two pieces in that ratio. Split $300 in 2:3? The pieces are A = 300 × 2/(2+3) = $120 and B = 300 × 3/(2+3) = $180. This is the bill-splitting / partnership-share / inheritance mode, and it generalizes immediately: each piece’s share equals its weight divided by the sum of the weights, multiplied by the total.

The Greatest Common Divisor (GCD) and Euclidean Algorithm

Simplifying a ratio means dividing both sides by the largest number that divides both of them evenly. That number is the greatest common divisor — sometimes called the highest common factor. For 12:18, the divisors of 12 are {1,2,3,4,6,12} and the divisors of 18 are {1,2,3,6,9,18}. The largest one shared by both sets is 6, so GCD(12,18) = 6 and the simplified ratio is 12/6 : 18/6 = 2:3.

Listing divisors works for small numbers but falls apart as soon as you have, say, GCD(2,310, 1,365). Euclid’s algorithm — written down around 300 BCE — solves the problem in a handful of steps no matter how large the numbers are. The trick is a single observation: GCD(a, b) = GCD(b, a mod b). Replace the larger number with the remainder when you divide it by the smaller, repeat until the remainder is 0, and the last non-zero number is your GCD.

Five steps for a problem that would take dozens of divisor lookups by hand. The algorithm is so efficient that it underpins modern cryptography (RSA key generation leans on a close cousin called the extended Euclidean algorithm), computer-algebra systems, and — yes — this little ratio calculator. Every “Simplify” result you see was produced by exactly this loop running on your inputs.

How to Use This Calculator

1. Pick a mode — Simplify, Equivalent, or Scale. The form fields adjust automatically; only the inputs that mode actually needs are shown.
2. Enter A and B — the two sides of your ratio. They must be positive whole numbers. If your raw values are decimals (like 1.5:2.25), multiply both sides by the smallest power of 10 that clears the decimals (here ×100 → 150:225) and the calculator will simplify from there.
3. For Equivalent mode, enter the target value — the new number on the A-side. The calculator returns the matching B-side.
4. For Scale mode, enter the total — the amount to be split. The calculator returns the two pieces and their decimal share, and you can verify they add back to your total.
5. Read the decimal value shown alongside the simplified form. Ratios and decimals are two views of the same number — A:B is just another way of writing the fraction A/B, so 2:3 = 0.667 and 16:9 ≈ 1.778. The decimal is what most computers and calculators want internally.

Three Worked Examples

Three concrete scenarios, one per mode. Try plugging them into the calculator above to confirm the numbers — they make the abstract rules above feel mechanical.

Example 1 — Simplify: 12:18

You measured a recipe at 12 grams of salt to 18 grams of sugar and want the cleanest way to write that down. Run Simplify mode with A = 12, B = 18.

• Compute GCD(12, 18). 18 mod 12 = 6; 12 mod 6 = 0; so GCD = 6.
• Divide both sides: 12 / 6 = 2 and 18 / 6 = 3.
• Simplified ratio: 2:3.
• Decimal: 2 / 3 ≈ 0.667.

That single result tells you a lot. Any time a recipe calls for the same ratio of salt to sugar — whether 2g:3g, 20g:30g, or 200g:300g — it’s the same blend. Simplification is what makes those equivalences obvious. For more on the underlying fraction-form math, the fraction calculator handles the addition, subtraction, and multiplication side of the same idea.

Example 2 — Equivalent: 2:3 with target 10

Your simplified recipe is 2:3 (salt:sugar). For a bigger batch you want to use 10 grams of salt. How much sugar keeps the flavor identical? Run Equivalent mode with A = 2, B = 3, target = 10.

• The unknown side is target × B / A = 10 × 3 / 2.
• 10 × 3 = 30. 30 / 2 = 15.
• Equivalent ratio: 2:3 = 10:15.

Sanity check by re-simplifying 10:15. GCD(10, 15) = 5; 10/5 : 15/5 = 2:3. Same ratio, just larger numbers — exactly what scaling a recipe should do. The same operation powers map distances (“1 cm on the map represents 50,000 cm on the ground; my route is 7 cm, how far is that in real life?”), currency conversion, and unit scaling. If you’re thinking in “parts per hundred” rather than parts per other-thing, the percentage calculator is the special case where the second side is fixed at 100.

Example 3 — Scale: split $300 in 2:3

Two business partners, Asad and Bilal, contributed unequally to a venture. Their agreed profit-share ratio is 2:3. The venture pays out $300. How much does each partner take home? Run Scale mode with A = 2, B = 3, total = 300.

• Total parts in the ratio: 2 + 3 = 5.
• Asad’s share: 300 × 2 / 5 = 600 / 5 = $120.
• Bilal’s share: 300 × 3 / 5 = 900 / 5 = $180.
• Verify: 120 + 180 = $300. ✓

Same arithmetic powers inheritance shares (Islamic and civil-law traditions both commonly express bequests in ratios), legal damages allocation, restaurant bill splits when not everyone ate equally, and dividend distribution among shareholders. The single rule is: each piece’s dollar value equals its weight in the ratio, divided by the sum of all weights, times the total being split.

Ratio vs Fraction vs Percentage

The three concepts are mathematically interchangeable but emphasize different things — getting fluent in switching between them is one of the most useful arithmetic skills.

• A ratiocompares two things directly. 2:3 means “for every 2 of the first, there are 3 of the second.” Both quantities are visible.
• A fractiontakes the same comparison and turns it into a single number — the share of one thing relative to a part or to the whole. 2:3 as a fraction-of-the-whole is 2/(2+3) = 2/5 (40%) for the first piece and 3/5 (60%) for the second. As a fraction-of-each-other it’s simply 2/3.
• A percentageis just a fraction whose denominator is fixed at 100. 2:3 → 2/5 = 40% for the first share. The fixed denominator is what makes percentages easy to compare at a glance — you can rank percentages directly, but two ratios with different totals can’t be eyeballed.

The right one to use depends on the question. Recipes and gear ratios stay as ratios. Inheritance shares get expressed as fractions (“one-fifth of the estate”). Tax rates, exam scores, and discount offers reach for percentages — see the discount calculator for the standard percentage-off pattern.

Common Mistakes

• Confusing “part to part” with “part to whole.” A 2:3 ratio means there are 5 total parts, not 3. So “2:3 of $300” is $120 + $180, not $200. The most-common miscalculation in inheritance and partnership splits — students who default to thinking 2:3 means “2 out of 3” will give one party $200 and the other $100 instead of the correct $120/$180.
• Adding ratios instead of recombining them.1:2 plus 1:3 is not 2:5. Ratios don’t add unless you first convert them to fractions of the same whole (1/3 + 1/4 = 7/12). If two paint mixtures are 1:2 and 1:3 and you combine equal volumes, the result’s ratio depends on the volumes mixed — not on the ratios themselves.
• Forgetting to simplify before comparing.Are 12:18 and 14:21 the same? Eyeballing says “maybe.” Simplifying says yes — both reduce to 2:3. Always simplify before deciding two ratios are different.
• Mixing up the order of A and B. Salt:sugar = 2:3 is not the same as sugar:salt = 2:3 — the second one means three times as much salt, not two-thirds as much. Always label which side is which when writing a ratio down, especially in recipes and chemistry.
• Using ratios with incompatible units. A ratio of 2 meters to 3 feet is nonsense unless one side is converted first. Convert both sides to the same unit before simplifying, or — even better — use unitless ratios where possible.
• Treating a ratio like a single number.2:3 is two pieces of information, not one. You can’t add 1 to a ratio meaningfully — you have to decide whether the 1 goes on the A-side, the B-side, or somehow both. Most ratio word-problem errors come from collapsing the two sides into one number too early.
• Rounding intermediate values. If you simplify 7:11 to a decimal (≈0.636) and then scale, small errors compound. Keep ratios in integer form as long as possible and round only the final answer.

When This Calculator Decides For You

Like most math tools, the calculator’s output usually maps to a real choice rather than a homework problem. The four most common ones:

1. Recipe scaling. You have a recipe for 4 people and 12 are coming. Run Equivalent mode with each ingredient and target = 12 / 4 × original quantity. Far fewer arithmetic mistakes than scaling each ingredient by hand, and the calculator keeps the fractions exact.
2. Partnership and inheritance splits. Anyone splitting a sum in a predetermined ratio — equity payouts, royalty distributions, sibling inheritance, freelancer-and-agent commissions — should run Scale mode with the agreed ratio and the actual amount. The verification step (pieces sum back to total) catches arithmetic errors before the money moves.
3. Mixing chemicals, paints, fertilizers, or cocktails. The bottle says 1:5 dilution; you have 2 cups of concentrate; how much water? Equivalent mode with A = 1, B = 5, target = 2 → answer: 10 cups water. Same procedure for fertilizer mixes and gardening — wrong dilution can scorch plants.
4. Comparing seemingly different rates. Two grocery items: one is $4 for 12 oz, another is $6 for 18 oz. Are they the same price per ounce? Simplify 12:4 vs 18:6 → both reduce to 3:1 oz/dollar. Identical unit price, despite the different sticker price. Simplification turns “is A or B a better deal” into a one-second comparison.

Pair this calculator with the percentage calculator when you want the share-of-whole view, and the GPA calculator when you’re combining weighted ratios across multiple items (a GPA is essentially a credit-weighted ratio of grade points to credit hours).

The Golden Ratio and Other Famous Ratios

Some ratios show up so often across nature, design, and engineering that they’ve earned their own names. The most famous is the golden ratio, denoted φ (phi) ≈ 1.618. It’s the unique ratio where dividing a line so the longer part is to the whole as the shorter part is to the longer — i.e., a:b = b:(a+b) — gives the same value. Architects from the Parthenon onward, Renaissance painters, and modern UI designers have all used φ to lay out pleasing proportions.

Other ratios you encounter constantly:

• 16:9 — the aspect ratio of nearly every modern screen, monitor, and streaming video. Decimal value 1.778. Replaced the older 4:3 standard around 2010 and almost no media is produced in anything else today.
• 3:2 — classic 35mm photographic film, still common on DSLR sensors and Apple/iPhone print sizes.
• √2 (≈ 1:1.414) — the ISO paper-size ratio used by A4, A3, A2 and their cousins. The ratio is preserved when you fold the paper in half, which is why A5 has the same proportions as A4.
• π ≈ 3.14159…— the ratio of any circle’s circumference to its diameter. Irrational, so it can’t be expressed exactly as a ratio of two whole numbers — but every approximation (22:7, 355:113) is a ratio in disguise.
• e ≈ 2.71828…— Euler’s number, the base of natural logarithms, arises in continuous compounding and exponential growth. Like π, it’s irrational but can be approximated by ratios as closely as needed.

The fact that any irrational number can be approximated arbitrarily well by simple ratios — a result called continued-fraction expansion— is one of the deeper ideas in number theory, and it’s why ratios remain relevant even when the underlying physical quantity isn’t a clean fraction.