Ratio Calculator

To reduce a ratio, divide both sides by their greatest common divisor. For 8:12 the GCD is 4, so it simplifies to 2:3 — the same relationship expressed in the smallest whole numbers. This is exactly how you reduce a fraction, because the ratio A:B and the fraction A/B carry identical information. The point of reducing is comparability. 8:12, 20:30 and 100:150 are indistinguishable as relationships, and only the reduced form makes that obvious at a glance. For awkward numbers, the Euclidean algorithm finds the GCD without any factoring: replace the larger number with its remainder against the smaller, and repeat until the remainder is zero. For 8 and 12: 12 mod 8 = 4, then 8 mod 4 = 0, so the GCD is 4. Two steps, and the method stays just as quick on numbers with hundreds of digits — which is why it is still in active use in cryptographic software. One rule that has nothing to do with arithmetic: both quantities must be in the same unit before you reduce. A ratio of 2 feet to 18 inches is not 2:18; convert first, and it is 24:18, which reduces to 4:3.

A proportion states that two ratios are equal: A:B = C:D. Know any three of the four and cross-multiplication delivers the last: D = B × C ÷ A. So 8:12 = 20:? gives 12 × 20 ÷ 8 = 30. Enter a value in the Scale A to field to solve it. Cross-multiplication is not a trick; it is one step of ordinary algebra. Writing the proportion as A/B = C/D and multiplying both sides by B × D clears both denominators at once and leaves A × D = B × C. Everything else is rearrangement. The part that actually needs care is the setup, not the solving. Both ratios must list their quantities in the same order. If the left side is miles per hour, the right side has to be miles per hour too — flipping one side inverts the answer, and the result will look perfectly plausible while being the reciprocal of the truth. The sanity check is direction. Scaling 8 up to 20 means growing by a factor of 2.5, so the partner must also grow: 12 becomes 30. If your answer moved the wrong way, the proportion was set up upside down.

This is the distinction that causes the most real-world errors, and it costs nothing to check. A ratio normally compares part to part. Mixing concrete at 2:3 means 2 units of one ingredient for every 3 of the other — and the batch is therefore 5 units in total. A fraction or a percentage compares part to whole, so the same mix is 2/5 (40%) of the first ingredient and 3/5 (60%) of the second. Calling that mix "40% to 60%" is right; calling it "2/3" is not, even though the ratio is written 2:3. The 2/3 figure is what you get by dividing one part by the other, which is a legitimate number — the decimal 0.667 that this calculator reports — but it answers a different question. So the same relationship appears as 2:3, as 2/5 and 3/5, as 40% and 60%, and as the decimal 0.667, all correct and none interchangeable. Before quoting any of them, decide whether the denominator you want is the other part or the whole.

A very common task is dividing a known total according to a ratio — shared costs, ingredient quantities, a bonus pool, an inheritance. The method is always the same three steps. Add the parts to get the total number of shares, divide the total by that number to find the value of one share, then multiply out. Splitting $600 in the ratio 2:3 gives 2 + 3 = 5 shares, so each share is 600 ÷ 5 = $120, and the split is $240 and $360. Ratios are not limited to two terms. A three-way split of 2:3:5 has 10 shares, so $600 divides into $120, $180 and $300. The procedure does not change. Always check the total. The parts must add back to the original amount — $240 + $360 = $600 — and if they do not, a rounding error or a mis-multiplied share is hiding somewhere. When the division is not exact, decide deliberately where the leftover cent goes rather than letting it disappear.

Once you recognise the form, ratios appear nearly everywhere a comparison is being made. Scale. A drawing at 1:50 means one unit on paper is 50 in reality. USGS topographic quadrangle maps are printed at 1:24,000, which is why one inch on those sheets represents 24,000 inches on the ground — exactly 2,000 feet. Aspect ratio. A 16:9 screen is 1.778 times wider than it is tall. Because a ratio ignores absolute size, that single figure describes a phone, a laptop and a cinema screen equally well. Unit rates. Miles per gallon, dollars per pound and words per minute are all ratios reduced so the second term is 1, which is what makes them comparable across different quantities. Reducing to a per-unit rate is the standard way to answer "which is the better deal". Mixes and dilutions. Fuel mixes, fertiliser concentrates and cocktail recipes are all part-to-part ratios, and scaling one up is precisely the proportion problem from earlier. The common thread is that a ratio of two quantities in the same unit is dimensionless — the units cancel. That is what lets you compare a scale model to a building, or a small batch to a large one, without anything else changing.