Multiplying Fractions

Key Terms

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

Multiplying Fractions at a Glance

Multiplying Fractions FAQ

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.

Key Terms

Fraction— the object being multiplied

Mixed Number— must be converted to improper form before multiplying

Reciprocal— multiplying by a reciprocal produces 1; connects multiplication to division

See All Arithmetic Definitions →

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

\frac{a}{b}

and

\frac{c}{d}

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

For example,

\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}

.

This rule makes sense geometrically. Taking

\frac{2}{3}

\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.

Two thirds of four fifths

The unit square is cut into 3 columns and 5 rows, 15 equal cells. Two of the three columns are 2/3 of the square; four of the five rows are 4/5 of it. The cells in both, amber, are 2/3 of 4/5: 2 · 4 = 8 of the 3 · 5 = 15 cells. That is why the numerators multiply and the denominators multiply: 2/3 × 4/5 = 8/15.

The product 8/15 is smaller than both factors, as it must be when both are less than 1.

Multiplying Fractions by Whole Numbers

A whole number can be written as a fraction with denominator 1. The number 5 is

\frac{5}{1}

. This allows the same multiplication rule to apply.

\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 \times \frac{2}{3}

means

\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 \times \frac{a}{b} = \frac{n \times a}{b}

Four times two thirds

4 × 2/3 is four copies of 2/3. Each copy shades 2 thirds, so together they shade 8 thirds. Eight thirds fill two whole bars and two thirds of a third bar: 4 × 2/3 = 8/3 = 2 2/3. The whole number multiplies the numerator; the size of the parts, the denominator, stays the same.

The same answer comes from the rule: 4/1 × 2/3 = 8/3.

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

\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:

\frac{4}{9} \times \frac{3}{8} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

Without canceling first, the product is

\frac{12}{72}

, which then requires finding the GCD 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

2\frac{1}{3} \times 1\frac{1}{2}

, first convert both to improper fractions. The value

2\frac{1}{3}

becomes

\frac{7}{3}

and

1\frac{1}{2}

becomes

\frac{3}{2}

\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

\frac{7}{1} \times \frac{1}{2} = \frac{7}{2}

immediately. See mixed numbers for conversion methods.

You can check any product of mixed numbers with the fraction calculator: it converts them to improper fractions and shows every step.

Multiplying More Than Two Fractions

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

\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

\frac{2}{5}

with no multiplication needed.

The Word "Of" Means Multiply

In mathematical contexts, the word "of" signals multiplication. Finding

\frac{2}{3}

of 12 means computing

\frac{2}{3} \times 12

\frac{2}{3} \times 12 = \frac{2 \times 12}{3} = \frac{24}{3} = 8

This interpretation appears constantly in applications. A recipe asks for

\frac{3}{4}

of a cup. A sale offers

\frac{1}{3}

off the original price. A problem states that

\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

\frac{n}{n}

for any nonzero

n

, 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

\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.

Multiplying Fractions at a Glance

The page covered fraction multiplication across several flavors — the basic rule, whole numbers, mixed numbers, multiple fractions in a row, the "of" convention, cross-canceling, and the three identity-style special cases involving 1, 0, and the reciprocal. The table below collects each situation with the move to make and a worked example.

Situation	What to do	Example
Fraction × fraction	multiply numerators, multiply denominators	(2⁄3)(4⁄5) = 8⁄15
Fraction × whole number	write the whole number over 1, or multiply only the numerator	(3⁄4) × 5 = 15⁄4 = 3 3⁄4
Mixed × mixed	convert each to an improper fraction first; never multiply whole and fractional parts separately	2 1⁄3 × 1 1⁄2 = (7⁄3)(3⁄2) = 7⁄2 = 3 1⁄2
Three or more fractions	multiply all numerators and all denominators; cancel across every pair before computing	(2⁄3)(3⁄4)(4⁄5) = 2⁄5
"of" in a phrase	translate "of" to ×	2⁄3 of 12 → (2⁄3) × 12 = 8
Before computing	cross-cancel any numerator with any denominator that share a common factor	(4⁄9)(3⁄8) → (1⁄3)(1⁄2) = 1⁄6
× 1	result is unchanged; basis for equivalent fractions via 1 = n⁄n	(2⁄3) × (5⁄5) = 10⁄15
× 0	result is 0	(5⁄7) × 0 = 0
× reciprocal	result is 1; foundation for dividing fractions	(3⁄4) × (4⁄3) = 12⁄12 = 1

Multiplying Fractions FAQ

Do you need a common denominator to multiply fractions?

No — that requirement belongs to addition and subtraction only. To multiply, go straight across: numerators together, denominators together. So 2/3 × 4/5 = 8/15 with no conversion step at all. The rule for adding fractions is the more demanding one, which is why many people carry it over to multiplication by mistake.Read more →

Why does multiplying fractions give a smaller result?

Because multiplying by a number less than 1 takes a part of something rather than repeating it. Finding 2/3 of 4/5 means subdividing a piece that is already smaller than a whole, so the product is smaller than either factor: 8/15. The pattern reverses once a factor exceeds 1 — multiplying by 3/2 makes the result larger.Read more →

Why must mixed numbers be converted before multiplying?

Because multiplying whole parts and fractional parts separately gives the wrong answer. For 2⅓ × 1½, pairing 2 × 1 with ⅓ × ½ would suggest 2⅙, but the true product is 3½. Convert each to an improper fraction first — 7/3 and 3/2 — then multiply across: 21/6 = 7/2 = 3½.Read more →