Ratio Calculator - Simplify, Equivalent Ratios, and Part-to-Whole

A ratio calculator is for relative size, not for a single fraction of a whole. 8:12 says “eight of these for every twelve of those.” Reducing it does not change the comparison: divide both parts by their GCD of 4 and you get 2:3, the default example on this page. The same GCD idea extends to three parts: gcd(a, b, c) divides every term.

Integers are the comfortable case. Decimals are allowed by scaling - 1.5:2.5 becomes 15:25, GCD 5, then 3:5. Parts must be positive; a zero part would make a GCD step and a split both degenerate. Nothing is uploaded. Inputs linger in localStorage on this browser for up to 30 days.

If you actually wanted a reduced fraction rather than a colon ratio, use the fraction calculator. If the GCD itself is the object of the exercise, the greatest common factor calculator isolates that step.

1. Simplify - enter a and b, optionally c. The result is lowest terms after scaling decimals.
2. Missing term - solve a:b = c:x with x = b × c / a. Cross-multiply to check: a x should equal b c.
3. Split - enter a total T. Each share is (part / sum of parts) × T. Two- or three-part ratios both work; leave c blank for two parts.
4. Copy the headline and the working lines, or clear every field. The split view is resizable on desktop.

Equivalent ratios are multiples of the same reduced pair. 2:3, 4:6, 8:12, and 16:24 all describe the same relative sizes. Algebraically, a:b equals c:d when a d = b c. Rearranged for a blank last term, x = b c / a.

Example: 2:5 = 8:x gives x = 5 × 8 / 2 = 20. Example with decimals: 1.5:4 = 6:x gives x = 16. If a is tiny, x blows up; that is proportional scaling, not a bug. If you need that missing value as a percent of a whole, the percentage calculator is the next tool, because a ratio is not automatically a percent.

Splitting 100 in 2:3 uses parts that sum to 5, so the shares are 40 and 60. Splitting 100 in 2:3:5 uses parts that sum to 10, so the shares are 20, 30, and 50. The shares always add back to T when the parts are positive and floating-point rounding is honest - the result panel prints a check sum.

This is where ratio and fraction part ways. 2:3 is not “two thirds of the money.” Two thirds of the money would be the first share only if the ratio were 2:1. The first person’s fraction of the whole is a / (a+b), which is 2/5 in the 2:3 example.

a:b:c is one comparison among three positive quantities, not two stacked two-part ratios (though you can recover a:b by dropping c). Simplify with a single GCD of all three integers. 6:9:15 divides by 3 to 2:3:5. Mixing units - mixing milliliters with grams without converting - makes a ratio meaningless even if the GCD step still runs.

Recipes, concrete mix, and alloy problems love three-part ratios. If a map says “sand : cement : gravel = 2:1:3” and you have 120 kg to mix, split mode with T = 120 is the direct computation. Leave c empty whenever the problem is only two groups.

A fraction a/b is part-to-whole when a is contained in b. A ratio a:b is part-to-part. A percent is a special part-to-whole with denominator 100. A rate often carries units (km/h) that a pure ratio has already canceled. Homework that says “express as a ratio in lowest terms” wants 2:3, not 2/3, even though those two strings are related.

Cross-multiplying to test equivalence is the two-term version of “same reduced form.” For three terms, reduce both triples and compare, or scale one triple until two of the three parts match and see whether the third agrees.

Word problems often give a:b and b:c separately. To merge them into a:b:c, scale so the shared part matches. If a:b = 2:3 and b:c = 6:5, rewrite 2:3 as 4:6 so the middle 6 agrees, then a:b:c = 4:6:5. Simplify if a GCD appears. This calculator will not stitch two ratios for you; type the already-aligned three parts into simplify or split.

The same scaling trick writes a missing-term problem that looks like “a is to b as c is to x” when a third quantity is involved. Keep the paired positions honest: the first term of the left ratio corresponds to the first term of the right. Swapping b and c by accident is how 15 becomes 6.4 with no algebra error except the setup.

A map scale 1:50 000 is a ratio of distances, already simplified. It is not a split of 50 001 length units. To convert 4 cm on the map, the missing-term view is 1:50000 = 4:x, so x = 200 000 cm = 2 km. Split mode would be the wrong button: you are not sharing 4 cm between “map” and “ground.”

Recipes are splits. A dough ratio flour:water:salt = 100:65:2 of a 1.67 kg batch is split mode with T = 1.67 (or 1670 g if you stay in grams). Keep the units of T the same as the shares you want out. Mixing paints 2:1 yellow to blue to make 300 mL is split 200 mL and 100 mL. If a tin already contains 80 mL of yellow and you must hit 2:1 after adding blue, that is a missing-term or algebra problem (blue = 40 mL), not a split of 300.

Concentration mixes (“how much 10% acid with 30% acid to make 20%”) are weighted averages, which can be written as an inverse ratio of the gaps: 10 parts of 30% with 10 parts of 10% for a 20% target in the equal-gap case, more generally alligation. This page will split a total in a:b once you have decided the parts; it will not derive the parts from percentages.

A unit rate puts one part equal to 1: 8:12 is 1 : 1.5 or 0.667 : 1 depending which side you normalize. Lowest-term integers 2:3 are usually clearer for homework. Engineering gear ratios sometimes keep a non-reduced form because the tooth counts are physical (48:36 teeth is a real mesh even though 4:3 is the reduced ratio). Simplifying tells you the angular-speed ratio; it does not change how many teeth you must cut.

Floating-point GCD after scaling can misbehave if you type 0.3333333 as a stand-in for 1/3. Prefer exact decimals or whole numbers. 1/3 : 1/2 should be entered as 2:3 after you clear the fractions yourself, or as 0.5:0.75 only if you accept a scaling GCD on the rounded decimals. Repeating decimals that you truncate at six places may simplify to a nearby ratio, not the exact fraction you had in mind.

Teachers sometimes ask for a ratio in the form 1:n. Take 2:3, divide both parts by 2, and get 1:1.5. That is equivalent but no longer all integers. This simplifier prefers integer lowest terms (2:3) because that is what “simplify the ratio” almost always means on a worksheet. If you need 1:n, divide the printed parts by a yourself after Copy.

Part-to-whole language uses “out of.” 2 out of 5 of the split 2:3 is the first share of the total, which is 40% of T, not the ratio 2:3. Writing 2:5 as a ratio of first-part to whole is valid but easy to mix with 2:3. Label the quantities: yellow:blue versus yellow:total. The calculator’s a, b, c boxes are always parts in a part-to-part colon, never “out of T” unless you are in split mode and T is the whole.

Negative ratios show up in directed segments on a number line, but they are excluded here. Zero parts would make a missing term blow up or a GCD ill-defined for “0:12.” If a mixture truly has none of one component, the ratio is not the right object - say “no yellow” instead of 0:1.

120 km in 2 hours is the ratio 120:2, which simplifies to 60:1 and is then read as 60 km per 1 hour. That last reading is a unit rate, not a split of 122 km. Use simplify to reduce 120:2, then interpret the 60 as kilometers and the 1 as hours. Currency “exchange 8:12” is the same pattern as the default 2:3: eight of one currency for twelve of the other, not a partition of twenty coins unless the problem says you are sharing a pile.

Aspect ratios for screens (16:9, 4:3) are simplified colon ratios of width to height. Scaling a 1920×1080 frame does not change 16:9. Split mode would be the wrong model unless you were allocating a fixed pixel budget between width and height under a ratio constraint - possible, but that is not how people usually use 16:9. Photographers crop to 3:2 or 1:1 the same way: simplify, then scale the long side until it matches the paper, rather than treating 5 inches as a total to share between width and height like a recipe split of a print budget.