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AD BC (Cross Multiplication Rule) — Definition, Formula & Examples

The AD BC rule (cross multiplication) is a shortcut for solving proportions: if two fractions are equal, you can multiply diagonally and set the products equal. Given ab=cd\frac{a}{b} = \frac{c}{d}, cross-multiplying gives a×d=b×ca \times d = b \times c.

For any proportion ab=cd\frac{a}{b} = \frac{c}{d} where b≠0b \neq 0 and d≠0d \neq 0, the equality holds if and only if ad=bcad = bc. This result follows from multiplying both sides of the equation by the product bdbd, eliminating both denominators simultaneously.

Key Formula

ab=cd  ⟹  ad=bc\frac{a}{b} = \frac{c}{d} \implies ad = bc
Where:
  • aa = Numerator of the first fraction
  • bb = Denominator of the first fraction (cannot be 0)
  • cc = Numerator of the second fraction
  • dd = Denominator of the second fraction (cannot be 0)

How It Works

Write your two equal fractions side by side: ab=cd\frac{a}{b} = \frac{c}{d}. Multiply the numerator of the first fraction by the denominator of the second to get adad. Multiply the denominator of the first fraction by the numerator of the second to get bcbc. Set these two products equal: ad=bcad = bc. Now solve the resulting equation for whichever variable is unknown. This technique works because you are really multiplying both sides by bdbd to clear the fractions.

Worked Example

Problem: Solve for x: 34=x20\frac{3}{4} = \frac{x}{20}
Cross-multiply: Multiply diagonally: the numerator of the left side times the denominator of the right side equals the denominator of the left side times the numerator of the right side.
3×20=4×x3 \times 20 = 4 \times x
Simplify: Compute the known product on the left side.
60=4x60 = 4x
Solve for x: Divide both sides by 4.
x=604=15x = \frac{60}{4} = 15
Answer: x=15x = 15

Why It Matters

Cross multiplication is the standard method for solving proportions in pre-algebra and algebra courses. You will use it frequently in unit conversions, scale drawings, similar triangles, and percent problems throughout middle school and beyond.

Common Mistakes

Mistake: Cross-multiplying when fractions are being added or subtracted instead of set equal.
Correction: Cross multiplication only applies to an equation where one fraction equals another (ab=cd\frac{a}{b} = \frac{c}{d}). For addition or subtraction of fractions, find a common denominator instead.