Proportions Explained: Ratios, Cross-Multiplication, and Percent Problems

A proportion is an equation between two ratios: a/b = c/d. It says the relationship between two quantities is the same as the relationship between two others. That simple claim does a surprising amount of work. It finds a missing value from three known ones, scales a recipe from 4 servings to 12, reads a map scale, and converts a ratio into a percentage. The method is cross-multiplication, and the discipline is pairing the quantities in the right order. This guide covers the method, the checks, and the cases where a proportion is the wrong tool.

What a proportion is, and what a ratio is

A ratio compares two quantities: 3 to 2, written 3:2 or 3/2. A proportion asserts that two ratios are equal: 3/6 = 9/18. The two ratios have the same value, 0.5, even though the numbers differ. Every proportion can be read as a statement about scaling: the second pair is the first pair multiplied by one factor. Here the factor is 3. Seeing the factor is often faster than cross-multiplying, and it is the check that catches pairing errors: if the factor does not apply consistently to both quantities, the proportion is set up wrong.

Ratios and fractions are the same numbers in different clothes, and a proportion can wear either. The ratio 3:2 is the fraction 3/2; the proportion 3:2 = 9:6 is the equation 3/2 = 9/6. Decimals and percentages are further outfits: 3/2 = 1.5 = 150%. The Proportion Calculator shows all four faces of an answer at once, which is useful precisely because school, work, and exam questions each prefer a different one.

Cross-multiplication, and checking the answer

Cross-multiplication follows from clearing the denominators. From a/b = c/d, multiply both sides by b × d and the fractions vanish, leaving a × d = b × c. To solve 3/6 = x/18: the known diagonal is 3 × 18 = 54, the other diagonal is 6 × x, so x = 54 ÷ 6 = 9. The rule never changes: the missing value is one diagonal product divided by the number opposite the missing value. It works with decimals and negatives too, though a negative in a proportion about physical quantities usually signals a setup error worth re-reading.

Check every answer by substitution, not by confidence. Put 9 back: 3/6 = 0.5 and 9/18 = 0.5. The two sides match, so the proportion holds. A second check is direction: if the quantity on one side grew, the paired quantity on the other side should have grown by the same factor. An answer larger than both known quantities on its side, when the other side shrank, is a sign the quantities were paired in the wrong order — the commonest proportion error, and the easiest to catch by asking whether the answer is sensible before moving on.

Percent problems are proportions in disguise

Every basic percent problem is the proportion part/whole = percent/100. “What percent of 80 is 20?” becomes 20/80 = p/100, so p = 20 × 100 ÷ 80 = 25. “What is 15% of 240?” becomes x/240 = 15/100, so x = 240 × 15 ÷ 100 = 36. “40 is 25% of what number?” becomes 40/w = 25/100, so w = 40 × 100 ÷ 25 = 160. One setup handles all three question types; the only skill is deciding which quantity is the part, which is the whole, and which is the percent.

Percent increase and decrease need one extra care. An increase of 20% on 50 is not 20% of the new value but of the old: x/50 = 120/100 gives 60. A decrease works the same way with 80/100. The base is always the starting value, never the ending one. This is where most percent errors live, and the proportion setup makes the base visible: it is the denominator on the left, and if the wrong number sits there, the setup itself shows it.

Direct and inverse proportion

In direct proportion, doubling one quantity doubles the other: twice the workers, twice the output, at the same rate. The proportion is a/b = c/d with quantities paired in the same order on both sides. In inverse proportion, doubling one quantity halves the other: twice the workers, half the time, for a fixed job. The setup flips: a/b = d/c, because the products, not the ratios, are constant. For a job taking 6 workers 8 hours, 12 workers take 6 × 8 ÷ 12 = 4 hours. Using the direct setup here would give 16 hours — more workers taking longer — which the sense check catches immediately.

Deciding which kind you have is the whole problem, and it is decided by the story, not by the numbers. Ask: if this quantity goes up, does the other go up too (direct), or down (inverse)? Speed and time for a fixed distance are inverse; distance and time at fixed speed are direct; price and quantity bought with fixed money are inverse; ingredients and servings are direct. If the answer is neither — the relationship is not a straight line through the origin — then no proportion setup is correct, and a different model is needed.

The unit-rate method, maps, and recipes

The unit-rate method finds the value of one unit first, then multiplies. If 5 kg costs $12.50, one kilogram costs $2.50, and 8 kg cost $20.00. It is often quicker than cross-multiplication for shopping and scaling, and it is the same mathematics: the unit rate is the common ratio, made visible. Use whichever method you can check. The unit rate is also the right tool when the question is “how much per one,” because it answers that question directly instead of burying it inside a proportion.

Map scales are proportions with units attached. A scale of 1:50,000 means 1 cm on the map stands for 50,000 cm — 500 m, or 0.5 km — on the ground. A 6 cm measurement on the map is therefore 6 × 0.5 = 3 km. The trap is the units: 1:50,000 in centimetres is not 1 cm to 50,000 metres. Convert the scale to one ground unit first, then multiply. Recipe scaling works the same way: the factor from 4 servings to 12 is 3, so every ingredient is multiplied by 3 — 200 g of flour becomes 600 g. The factor must be applied to every ingredient, including salt and leavening, or the recipe’s balance changes with its size.

Proportions that are not proportions deserve a warning of their own. The ratio of two people’s ages changes over time: a father of 36 and a son of 12 are in ratio 3:1, but 10 years later they are 46 and 22, a ratio of 23:11, because the same 10 years were added to both. Scaling the ratio would wrongly predict a son of 15. Age problems are solved by adding the years to each age separately, never by scaling. Any quantity that grows by addition rather than multiplication breaks the proportion model, and recognising that is more valuable than any amount of cross-multiplication.

Similar figures: the scale factor, squared and cubed

When two figures are similar — the same shape, different size — every length in one is the corresponding length in the other multiplied by one scale factor. A photograph enlarged by a factor of 2 is twice as wide and twice as tall. But its area is multiplied by the square of the factor, 4, and if it were a solid, its volume would be multiplied by the cube, 8. The length, area, and volume scale differently, and confusing them is the classic error of scale: doubling the side of a square quadruples its area, and doubling the side of a cube multiplies its volume by eight.

The area rule is the one that bites in daily life. A pizza of 12 inches diameter has an area of about 113 square inches; a pizza of 18 inches has an area of about 254 square inches — 2.25 times as much, because (18 ÷ 12)² = 2.25 — while costing nothing like 2.25 times as much, which is why the larger pizza is usually the better value. Paint, carpet, and turf follow the same rule: the cost of covering a floor twice as long and twice as wide is four times, not twice, and a quote that scales it by two has underpriced the job by half.

Maps and models use the scale factor directly. A model built at 1:72 scale has every length 1/72 of the original; its surface area is 1/5,184 of the original’s, and its volume 1/373,248. A scale drawing that is right in its lengths is right in its proportions, and the area and volume follow by squaring and cubing — they are not separate measurements to be checked, but consequences of the one factor. The proportion setup handles all of it: write the ratio of the lengths, and the ratio of the areas is the square of that ratio, whatever the shapes.

Frequently asked questions