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Dividing Fractions



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



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.



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 GCF. 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.