Cross Multiplication Calculator

Cross-multiplication is the standard algebraic technique for solving proportions, equations that set two ratios equal to each other. It is one of the most versatile tools in applied mathematics.

The Proportion Setup:

a/b = c/d

Cross-Multiplication Rule:

a × d = b × c

To solve for any one unknown, isolate it:

• Solve for a: a = (b × c) / d
• Solve for b: b = (a × d) / c
• Solve for x in a/b = x/d: x = (a × d) / b

Worked Example 1, finding an unknown: If 3 apples cost $1.50, how much do 7 apples cost?

• Setup: 3/1.50 = 7/x
• Cross-multiply: 3x = 1.50 × 7 = 10.50
• x = 10.50 / 3 = $3.50

Worked Example 2, scaling a recipe: A recipe for 4 servings uses 2.5 cups of flour. How much for 10 servings?

• Setup: 4/2.5 = 10/x
• Cross-multiply: 4x = 2.5 × 10 = 25
• x = 25 / 4 = 6.25 cups

Worked Example 3, percentage problems: 35 is what percent of 140?

• Setup: 35/140 = x/100
• Cross-multiply: 140x = 3,500
• x = 25%

Getting the setup right is the whole game.

The arithmetic here is trivial. Almost every wrong answer comes from writing the proportion down badly, and there is one rule that prevents it: keep like units in like positions. If the top of the left ratio is apples, the top of the right ratio must also be apples. Mix them, as in 3 apples / $1.50 = $x / 7 apples, and you get a number that is not wrong so much as meaningless.

A useful check once you have an answer: read the two ratios back as sentences. “3 apples cost $1.50” and “7 apples cost $3.50” both come out to $0.50 an apple, so the setup held. If the two sentences describe different rates, the proportion was built wrong.

The rearrangements are all equivalent. From a/b = c/d you can also write a/c = b/d, or b/a = d/c, or d/b = c/a. All four cross-multiply to the same ad = bc, so pick whichever arrangement makes the unknown easiest to isolate. Textbooks call this the means-extremes property, which is a grander name than it deserves.

When it does not apply. Cross-multiplication assumes direct proportion: double one side and the other doubles. Plenty of real relationships are inverse instead. Four painters take 6 days, so eight painters take 3, not 12. That is x·y = k, not x/y = k, and setting it up as a proportion gives you an answer twice as wrong as doing nothing. Ask first whether the quantities rise together or trade off against each other.

Real-world uses: unit conversions, map scaling, medication dosing, recipe scaling, currency exchange, grade weighting, and any time you need to hold a ratio steady while one value changes. Cross-multiplication is the algebraic backbone of direct proportionality.