Ratio / Proportion Solver

A proportion states that two ratios are equal. Given three of the four terms, the fourth follows from cross multiplication. It is the most broadly useful piece of arithmetic there is, and it turns up in scaling recipes, reading map distances, mixing chemicals, and converting units.

How to use it

1. Enter the three known values as a, b, and c.
2. The solver computes x such that a : b equals c : x.
3. Read the missing value and the completed proportion.

Cross multiplication

If a divided by b equals c divided by x, then multiplying both sides by b times x gives a times x equals b times c, so x equals b times c divided by a.

A worked example: 2 is to 3 as 10 is to what. x equals 3 times 10 divided by 2, which is 15. Checking, 2/3 and 10/15 both reduce to 2/3.

The one thing to get right is which position the unknown occupies, because the arrangement changes the arithmetic. Keeping units in the same position on both sides prevents this: if the left ratio is miles per hour, the right must also be miles per hour, not hours per mile. Writing the units alongside the numbers catches inverted setups immediately.

Scaling recipes and drawings

Recipe scaling is straightforwardly proportional for ingredients and not proportional for anything else. Doubling a recipe doubles the flour and the sugar. It does not double the baking time, the pan size, or the oven temperature.

Cooking time scales with thickness rather than volume, roughly as the square root, because heat has to conduct to the centre. Doubling a cake batter into the same-sized pan gives a much thicker cake needing considerably longer at a slightly lower temperature. Doubling into two pans keeps the thickness and the time roughly unchanged. Salt and leavening also often need less than a proportional increase, particularly at large multiples.

Scale drawings are properly proportional in one dimension. A 1:50 drawing means one unit on paper is fifty in reality, so a 40 mm line is 2,000 mm — two metres. What is not proportional is area or volume: doubling every linear dimension quadruples the area and multiplies the volume by eight. Estimating paint or concrete from a linear scale factor is a common and expensive error.

Ratios with more than two terms

Mixing ratios are often written with three or more parts — 1:2:3 for cement, sand, and aggregate, for instance. The method is to total the parts and divide the target quantity by that total to get the value of one part.

For 1:2:3 totalling six parts across 12 cubic metres, one part is 2 cubic metres, giving 2, 4, and 6. The advantage of working in parts is that the proportion holds at any total, so the same specification works for a wheelbarrow and a truckload.

A frequent confusion is ratio against fraction. A 1:3 ratio means one part to three parts, so the first component is one quarter of the whole and not one third. Written as a fraction of the total it is 1/4. Dilution instructions such as "dilute 1:10" are ambiguous in ordinary use and can mean one part concentrate to ten parts water, giving eleven parts total, or one part making up ten parts total. For chemicals where the difference matters, the product documentation is worth checking rather than guessing.

When the relationship is not proportional

Proportional reasoning assumes a straight line through the origin: doubling the input doubles the output, and zero input gives zero output. A great many real relationships fail one or both conditions.

Anything with a fixed component fails the origin test. A taxi fare with a flag-fall charge, a utility bill with a standing charge, or a job with a setup cost is linear but not proportional, and scaling it proportionally overcharges short trips and undercharges long ones.

Then there are relationships that are not linear at all. Drug dosing by body weight breaks down at the extremes and is often calculated by body surface area instead. Wind resistance rises with the square of speed. Beam strength depends on depth cubed. Labour on a project does not halve when you double the team.

The check is to ask whether zero input plausibly gives zero output, and whether the ratio between two known pairs is actually the same. If a 10-unit order costs $100 and a 20-unit order costs $180, the relationship is not proportional and setting up a proportion will give a wrong answer confidently.

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Everyday math — A 50 percent gain and a 50 percent loss leave you down 25 percent, and that is the least surprising thing here.

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