A proportion says that two ratios describe the same relationship: a/b = c/d. If three of those four numbers are known, the fourth is fixed — there is exactly one value that keeps both sides equal. That single idea is what lets you scale a recipe from four servings to seven, convert a 1:50 architectural drawing into real metres, work out how much paint a wall twice the size needs, or read a currency rate as a ratio rather than a lookup table. Enter the three values you have below and the calculator returns the missing one together with the cross-multiplication step, so you can check the arithmetic instead of trusting a black box.
**How cross-multiplication works.** Starting from a/b = c/d, multiply both sides by b and by d. The denominators cancel and you are left with a × d = b × c. From there you divide by whichever number sits next to the unknown. Solving for d gives d = (b × c) / a; solving for a gives a = (b × c) / d; solving for b gives b = (a × d) / c; solving for c gives c = (a × d) / b. Every proportion problem is one of those four rearrangements — there is nothing else to memorise.
**Keep the units in the same position.** The most common mistake is mixing the order of the terms. If the left ratio is *miles per hour*, the right ratio must also be *miles per hour*, not *hours per mile*. Write the units next to the numbers on your first attempt: 3 cups / 4 servings = x cups / 7 servings. When the same unit appears on top of both fractions and the other unit on the bottom of both, the answer comes out in the unit you expected.
**Direct versus inverse relationships.** Cross-multiplication assumes the two quantities move together: double one and the other doubles. That is true for ingredients, map scales, unit prices, and mixing ratios. It is *not* true for things like workers versus time to finish a job, where doubling the workers halves the duration. Those are inverse proportions and follow a × b = c × d instead. If your answer moves in the wrong direction, you have almost certainly applied the direct form to an inverse problem.
Proportion (cross-multiplication) formula
a / b = c / d → a × d = b × c → x = (product of the two knowns diagonal to x) ÷ (the known beside x)
A proportion is solved by cross-multiplication. Multiplying both sides of a/b = c/d by b·d clears the denominators and leaves a × d = b × c. Whichever term is unknown, you multiply the two values that sit diagonally opposite it and divide by the value directly beside it. So d = (b × c) / a, c = (a × d) / b, b = (a × d) / c, and a = (b × c) / d. The result is exact — no rounding is involved beyond how you choose to display it.
A recipe uses 3 cups of flour for 4 servings. You are cooking for 7 people and want to know how much flour that takes.
Result: 5.25 cups of flour (5 cups plus 4 tablespoons)
Cross-multiply so that a × d = b × c, then divide by the number sitting next to the unknown. For example, in 3/4 = x/7 you get 3 × 7 = 4x, so x = 21/4 = 5.25.
It removes the fractions. Multiplying both sides of a/b = c/d by b and by d cancels both denominators and leaves the equivalent equation a × d = b × c, which is a single multiplication you can solve directly.
A zero is fine in a numerator — 0/5 = 0/9 is a valid statement. A zero in a denominator is not, because division by zero is undefined, and a zero in the position next to the unknown makes the problem unsolvable since you would have to divide by it.
When the two quantities are inversely related. If four painters finish a room in 6 hours, eight painters do not need 12 hours. Inverse problems use a × b = c × d, so 4 × 6 = 8 × x gives x = 3 hours.
One unit on the drawing represents 50 of the same units in reality. Set it up as 1/50 = measured/actual, keeping both measurements in the same unit, then convert the answer at the end if you want metres or feet.
Divide each side out as a decimal. Both fractions must give the same value — 3/4 = 0.75 and 5.25/7 = 0.75. If the two decimals differ, one of the four numbers is in the wrong position.