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Multiplying Fractions



Multiplying Fractions — Basic Rule
Multiplying Fractions by Whole Numbers
Cross-Canceling Before Multiplying
Multiplying Mixed Numbers
Multiplying More Than Two Fractions
The Word "Of" Means Multiply
Special Cases



Numerator Times Numerator, Denominator Times Denominator

Multiplying fractions is more straightforward than adding or subtracting because no common denominator is required. The rule is direct: multiply the numerators together and multiply the denominators together. This page covers fraction-by-fraction multiplication, multiplying fractions by whole numbers, cross-canceling to simplify, and handling mixed numbers.



Multiplying Fractions — Basic Rule

To multiply two fractions, multiply the numerators to get the new numerator and multiply the denominators to get the new denominator. For fractions ab\frac{a}{b} and cd\frac{c}{d}:

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}


For example, 23×45=2×43×5=815\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}.

This rule makes sense geometrically. Taking 23\frac{2}{3} of 45\frac{4}{5} means subdividing something already divided into fifths, then taking a portion. The product describes the fraction of the original whole that remains.

Multiplying Fractions by Whole Numbers

A whole number can be written as a fraction with denominator 1. The number 5 is 51\frac{5}{1}. This allows the same multiplication rule to apply.

34×5=34×51=154=334\frac{3}{4} \times 5 = \frac{3}{4} \times \frac{5}{1} = \frac{15}{4} = 3\frac{3}{4}


Alternatively, think of multiplying a fraction by a whole number as repeated addition. The expression 4×234 \times \frac{2}{3} means 23+23+23+23=83\frac{2}{3} + \frac{2}{3} + \frac{2}{3} + \frac{2}{3} = \frac{8}{3}.

Both interpretations yield the same result. The whole number multiplies the numerator while the denominator remains unchanged: n×ab=n×abn \times \frac{a}{b} = \frac{n \times a}{b}.

Cross-Canceling Before Multiplying

Simplifying before multiplying keeps numbers smaller and avoids reducing large products afterward. Cross-canceling divides any numerator and any denominator by their common factor before computing the product.

Consider 49×38\frac{4}{9} \times \frac{3}{8}. The numerator 4 and denominator 8 share a factor of 4. The numerator 3 and denominator 9 share a factor of 3. Canceling both:

49×38=13×12=16\frac{4}{9} \times \frac{3}{8} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}


Without canceling first, the product is 1272\frac{12}{72}, which then requires finding the GCF to reduce. Cross-canceling reaches the simplified answer directly.

Multiplying Mixed Numbers

Mixed numbers must be converted to improper fractions before multiplying. Attempting to multiply whole parts and fractional parts separately produces incorrect results.

To compute 213×1122\frac{1}{3} \times 1\frac{1}{2}, first convert both to improper fractions. The value 2132\frac{1}{3} becomes 73\frac{7}{3} and 1121\frac{1}{2} becomes 32\frac{3}{2}.

73×32=216=72=312\frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2}


Cross-canceling applies here too. The 3 in the numerator and the 3 in the denominator cancel, leaving 71×12=72\frac{7}{1} \times \frac{1}{2} = \frac{7}{2} immediately. See mixed numbers for conversion methods.

Multiplying More Than Two Fractions

The multiplication rule extends to any number of fractions. Multiply all numerators together and all denominators together.

23×34×45=2×3×43×4×5=2460=25\frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} = \frac{2 \times 3 \times 4}{3 \times 4 \times 5} = \frac{24}{60} = \frac{2}{5}


Cross-canceling across all fractions before multiplying is particularly efficient here. In the example above, the 3 in the first denominator cancels with the 3 in the second numerator; the 4 in the second denominator cancels with the 4 in the third numerator. What remains is 25\frac{2}{5} with no multiplication needed.

The Word "Of" Means Multiply

In mathematical contexts, the word "of" signals multiplication. Finding 23\frac{2}{3} of 12 means computing 23×12\frac{2}{3} \times 12.

23×12=2×123=243=8\frac{2}{3} \times 12 = \frac{2 \times 12}{3} = \frac{24}{3} = 8


This interpretation appears constantly in applications. A recipe asks for 34\frac{3}{4} of a cup. A sale offers 13\frac{1}{3} off the original price. A problem states that 25\frac{2}{5} of the students are absent. Each "of" translates to multiplication.

Special Cases

Multiplying by 1 leaves a fraction unchanged. Since 1 equals nn\frac{n}{n} for any nonzero nn, this is the basis for creating equivalent fractions.

Multiplying by 0 produces 0. Any fraction times zero equals zero because the numerator of the product is zero.

Multiplying by a fraction's reciprocal produces 1. The product 34×43=1212=1\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1. This property is central to dividing fractions, where division is recast as multiplication by the reciprocal.