Key takeaway: Cross multiplication is a shortcut for solving a proportion by multiplying the numerator of each fraction by the denominator of the other and setting the two products equal.
Grade 77.RP.A.2cRatios, Rates and Percents
Definition of Cross Multiplication
Cross multiplication turns a proportion with fractions into an equation without them. For the proportion \(\frac{5}{8} = \frac{n}{36}\), multiply the top of the left fraction by the bottom of the right one, and the bottom of the left by the top of the right. The two products are equal: \(5 \times 36 = 8 \times n\).
It works because multiplying both sides of the proportion by both denominators clears the fractions in one move. After the crossing, you have a one-step equation: divide by the number multiplying the letter. The method applies only to an equation with exactly one fraction on each side.
Formula and notation
For nonzero \(b\) and \(d\):
\[\frac{a}{b} = \frac{c}{d} \quad\text{means}\quad a \cdot d = b \cdot c\]
The two products form an X when drawn on the fractions. If one term is unknown, solve the resulting equation for it.
Key facts
- The diagonal products of a true proportion are always equal.
- Cross multiplication only works when each side is a single fraction.
- After crossing, divide by the number that multiplies the unknown.
- Check the answer by simplifying both original fractions.
Cross Multiplication example
A recipe uses 5 ounces of juice concentrate for every 8 cups of water. How many ounces of concentrate go with 36 cups of water?
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Set up with concentrate on top: \(\frac{5}{8} = \frac{n}{36}\).
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Cross multiply: \(5 \times 36 = 8 \times n\), so \(180 = 8n\).
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Divide both sides by 8: \(n = \frac{180}{8} = 22.5\).
The batch needs 22.5 ounces of concentrate.
Common mistake
Watch out
Wes multiplies straight across instead of diagonally: \(5 \times n = 8 \times 36\), getting \(n = 57.6\).
Why it is wrong
That pairs the numerators and the denominators on the same side of the X, which is not how the diagonals work. The unknown must be multiplied by the opposite denominator: \(8n = 5 \times 36\), so \(n = 22.5\). The wrong answer 57.6 is more concentrate than the whole recipe could need.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Ratios, Rates and Percents