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Dividing Fractions



Key Terms
Reciprocals
Dividing Fractions — Basic Rule
Why Multiply by the Reciprocal
Dividing Fractions by Whole Numbers
Dividing Whole Numbers by Fractions
Dividing Mixed Numbers
Simplifying and Cross-Canceling
Word Problems and Applications
Dividing Fractions at a Glance



Multiply by the Reciprocal

Dividing fractions transforms into multiplication through the use of reciprocals. The rule "keep, change, flip" captures the process: keep the first fraction, change division to multiplication, and flip the second fraction. This method applies to all cases involving fractions, whole numbers, and mixed numbers.

Key Terms

Reciprocal— defined and used here; dividing by a fraction means multiplying by its reciprocal
Fraction— the object being divided
Mixed Number— must be converted to improper form before dividing

See All Arithmetic Definitions →


Reciprocals

The reciprocal of a fraction is obtained by swapping its numerator and denominator. The reciprocal of 34\frac{3}{4} is 43\frac{4}{3}. The reciprocal of 72\frac{7}{2} is 27\frac{2}{7}.

A number multiplied by its reciprocal always equals 1. The product 34×43=1212=1\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1 demonstrates this property.

Whole numbers have reciprocals too. Since 5 equals 51\frac{5}{1}, its reciprocal is 15\frac{1}{5}. The product 5×15=15 \times \frac{1}{5} = 1 confirms the relationship.

Zero has no reciprocal. Flipping 01\frac{0}{1} produces 10\frac{1}{0}, which is undefined. This reflects the impossibility of dividing by zero.

Dividing Fractions — Basic Rule

To divide one fraction by another, multiply the first fraction by the reciprocal of the second. For fractions ab\frac{a}{b} and cd\frac{c}{d}:

ab÷cd=ab×dc=a×db×c\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}


For example, 23÷45=23×54=1012=56\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}.

The mnemonic "keep, change, flip" summarizes the steps: keep the first fraction as is, change division to multiplication, flip the second fraction to its reciprocal.

Why Multiply by the Reciprocal

Division asks how many times one quantity fits into another. The expression 12÷14\frac{1}{2} \div \frac{1}{4} asks how many quarters fit into one half. Since two quarters make a half, the answer is 2.

Multiplying by the reciprocal produces this result: 12×41=42=2\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2.

The method works because multiplying by a reciprocal undoes multiplication. If ab×cd=x\frac{a}{b} \times \frac{c}{d} = x, then x÷cd=x×dcx \div \frac{c}{d} = x \times \frac{d}{c} returns to ab\frac{a}{b}. Division and multiplication by reciprocal are inverse operations.

Dividing Fractions by Whole Numbers

A whole number can be written as a fraction with denominator 1. To divide a fraction by a whole number, multiply by the reciprocal of that whole number.

34÷2=34÷21=34×12=38\frac{3}{4} \div 2 = \frac{3}{4} \div \frac{2}{1} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}


The result is smaller than the original fraction, which makes sense: dividing something into more parts yields smaller pieces.

Another example: 56÷3=56×13=518\frac{5}{6} \div 3 = \frac{5}{6} \times \frac{1}{3} = \frac{5}{18}.

Dividing Whole Numbers by Fractions

Dividing a whole number by a fraction produces a result larger than the original whole number when the fraction is less than 1. This reflects the question: how many small pieces fit into the whole?

6÷12=6×21=126 \div \frac{1}{2} = 6 \times \frac{2}{1} = 12


Six wholes contain twelve halves. The reciprocal method captures this directly.

Another example: 4÷23=4×32=122=64 \div \frac{2}{3} = 4 \times \frac{3}{2} = \frac{12}{2} = 6. Four wholes contain six two-thirds.

Dividing Mixed Numbers

Mixed numbers must be converted to improper fractions before dividing. The conversion methods are detailed on the mixed numbers page.

To compute 312÷1343\frac{1}{2} \div 1\frac{3}{4}, first convert both values. The mixed number 3123\frac{1}{2} becomes 72\frac{7}{2} and 1341\frac{3}{4} becomes 74\frac{7}{4}.

72÷74=72×47=2814=2\frac{7}{2} \div \frac{7}{4} = \frac{7}{2} \times \frac{4}{7} = \frac{28}{14} = 2


Cross-canceling before multiplying simplifies the work. The 7s cancel, leaving 12×41=2\frac{1}{2} \times \frac{4}{1} = 2.

Simplifying and Cross-Canceling

After flipping the second fraction, cross-cancel common factors before multiplying. This is identical to the technique used when multiplying fractions.

For 89÷43\frac{8}{9} \div \frac{4}{3}, rewrite as 89×34\frac{8}{9} \times \frac{3}{4}. The 8 and 4 share a factor of 4, and the 9 and 3 share a factor of 3:

89×34=23×11=23\frac{8}{9} \times \frac{3}{4} = \frac{2}{3} \times \frac{1}{1} = \frac{2}{3}


Without cross-canceling, the product 2436\frac{24}{36} requires reduction using the GCD. Canceling first avoids larger numbers entirely.

Word Problems and Applications

Division by a fraction answers "how many groups of this size fit into that amount?" The expression 34÷14\frac{3}{4} \div \frac{1}{4} asks how many quarters are in three-quarters. The answer is 3.

Practical applications include portioning and measurement. If a recipe uses 23\frac{2}{3} cup of flour per batch, and you have 4 cups, how many batches can you make? Compute 4÷23=4×32=64 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6 batches.

Rate problems also use fraction division. If a pipe fills 14\frac{1}{4} of a tank in 12\frac{1}{2} hour, the rate is 14÷12=14×2=12\frac{1}{4} \div \frac{1}{2} = \frac{1}{4} \times 2 = \frac{1}{2} tank per hour.

Dividing Fractions at a Glance

The page covered fraction division across every input flavor — basic rule, whole numbers in either position, mixed numbers, cross-canceling, the reciprocal concept that powers the rule, and the "how many fit" interpretation behind word problems. The table below collects each situation with the move to make and a worked example.
Situation What to do Example
Reciprocal of a fraction swap numerator and denominator; the product with the original is always 1 recip of 3⁄4 is 4⁄3; (3⁄4)(4⁄3) = 1
Zero has no reciprocal flipping 0⁄1 gives 1⁄0, which is undefined cannot divide by 0
Fraction ÷ fraction keep, change, flip — multiply the first by the reciprocal of the second (2⁄3) ÷ (4⁄5) = (2⁄3)(5⁄4) = 5⁄6
Fraction ÷ whole number write the whole over 1, then multiply by its reciprocal; result is smaller than the original (3⁄4) ÷ 2 = (3⁄4)(1⁄2) = 3⁄8
Whole number ÷ fraction multiply the whole by the reciprocal of the fraction; if the fraction is < 1, result is larger than the whole 6 ÷ (1⁄2) = 6 × 2 = 12
Mixed ÷ mixed convert both to improper fractions first, then keep-change-flip 3 1⁄2 ÷ 1 3⁄4 = (7⁄2)(4⁄7) = 2
After flipping cross-cancel any numerator with any denominator that share a common factor before multiplying (8⁄9)(3⁄4) → (2⁄3)(1⁄1) = 2⁄3
Why it works multiplication by a reciprocal undoes multiplication; division and ×-by-reciprocal are inverse operations x ÷ (c⁄d) = x × (d⁄c)
"How many fit" meaning a ÷ b answers how many copies of b fit into a — the basis of most word problems (3⁄4) ÷ (1⁄4) = 3 quarters fit in 3⁄4
Word-problem template amount ÷ size-per-group → number of groups 4 cups ÷ (2⁄3) cup per batch = 6 batches