Why do you flip the second fraction when dividing?

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of $\frac{c}{d}$ is $\frac{d}{c}$ because $\frac{c}{d} \times \frac{d}{c} = 1$. So flipping the second fraction converts the division problem into an equivalent multiplication problem, which is easier to carry out.

How do you divide a whole number by a fraction?

Write the whole number as a fraction over 1. For example, $6 \div \frac{3}{4}$ becomes $\frac{6}{1} \div \frac{3}{4}$. Then apply keep-change-flip: $\frac{6}{1} \times \frac{4}{3} = \frac{24}{3} = 8$. This makes sense because you are asking how many groups of $\frac{3}{4}$ fit into 6.

Can you divide fractions with different denominators?

Yes. Unlike adding or subtracting fractions, you do not need a common denominator to divide. Just use keep-change-flip and multiply straight across. The denominators take care of themselves in the multiplication.

Dividing Fractions — Definition, Formula & Examples

Dividing fractions is the process of finding how many times one fraction fits into another. You divide fractions by multiplying the first fraction by the reciprocal (flipped version) of the second fraction.

For any two fractions $\frac{a}{b}$ and $\frac{c}{d}$ , where $b \neq 0$ , $c \neq 0$ , and $d \neq 0$ , the quotient $\frac{a}{b} \div \frac{c}{d}$ is defined as $\frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$ . This operation is equivalent to multiplying the dividend by the multiplicative inverse of the divisor.

Key Formula

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

Where:

$a$ = Numerator of the first fraction (dividend)
$b$ = Denominator of the first fraction (dividend), must not be zero
$c$ = Numerator of the second fraction (divisor), must not be zero
$d$ = Denominator of the second fraction (divisor), must not be zero

How It Works

The method is often remembered as "keep, change, flip." Keep the first fraction exactly as it is. Change the division sign to a multiplication sign. Flip the second fraction (swap its numerator and denominator) to get its reciprocal. Then multiply the two numerators together and the two denominators together. Finally, simplify the resulting fraction if possible by dividing the numerator and denominator by their greatest common factor.

Worked Example

Problem: Compute

\frac{3}{4} \div \frac{2}{5}

Keep the first fraction: Write down the first fraction unchanged.

\frac{3}{4}

Change division to multiplication: Replace the ÷ sign with a × sign.

\frac{3}{4} \times

Flip the second fraction: The reciprocal of

\frac{2}{5}

\frac{5}{2}

\frac{3}{4} \times \frac{5}{2}

Multiply across: Multiply the numerators:

3 \times 5 = 15

. Multiply the denominators:

4 \times 2 = 8

\frac{15}{8}

Simplify: 15 and 8 share no common factors other than 1, so the fraction is already in simplest form. As a mixed number, this equals

1\frac{7}{8}

\frac{15}{8} = 1\frac{7}{8}

Answer:

\frac{3}{4} \div \frac{2}{5} = \frac{15}{8} = 1\frac{7}{8}

Another Example

This example involves a mixed number that must be converted first, and it shows how cross-canceling before multiplying keeps the numbers small.

Problem: Compute

2\frac{1}{3} \div \frac{7}{6}

Convert the mixed number to an improper fraction: Multiply the whole number by the denominator, then add the numerator:

2 \times 3 + 1 = 7

2\frac{1}{3} = \frac{7}{3}

Set up keep-change-flip: Keep

\frac{7}{3}

, change ÷ to ×, and flip

\frac{7}{6}

to get

\frac{6}{7}

\frac{7}{3} \times \frac{6}{7}

Cancel common factors before multiplying: The 7 in the numerator of the first fraction and the 7 in the denominator of the second fraction cancel.

\frac{\cancel{7}}{3} \times \frac{6}{\cancel{7}} = \frac{1}{3} \times \frac{6}{1}

Multiply and simplify: Multiply across:

\frac{1 \times 6}{3 \times 1} = \frac{6}{3} = 2

\frac{6}{3} = 2

Answer:

2\frac{1}{3} \div \frac{7}{6} = 2

Why It Matters

Dividing fractions appears throughout pre-algebra and algebra, from solving equations like

\frac{2}{3}x = 4

to working with rates such as miles per half-hour. Careers in cooking, construction, and nursing regularly require dividing fractional measurements — for instance, splitting

\frac{3}{4}

of a cup of flour into equal portions.

Common Mistakes

Mistake: Flipping the first fraction instead of the second

Correction: Always keep the first fraction (the dividend) the same. Only the second fraction (the divisor) gets flipped to its reciprocal.

Mistake: Trying to find a common denominator before dividing

Correction: A common denominator is needed for adding and subtracting fractions, not for dividing. Just use keep-change-flip and multiply across.

Mistake: Forgetting to convert mixed numbers to improper fractions

Correction: The keep-change-flip method works on fractions, not mixed numbers. Always convert mixed numbers first, then divide.

Check Your Understanding

Compute

\frac{5}{6} \div \frac{2}{3}

Hint: Flip

\frac{2}{3}

to get

\frac{3}{2}

, then multiply. Simplify at the end.

Answer:

\frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}

Compute

4 \div \frac{1}{2}

Hint: Write 4 as

\frac{4}{1}

first, then flip and multiply.

Answer:

\frac{4}{1} \times \frac{2}{1} = 8

Compute

1\frac{1}{2} \div 3\frac{3}{4}

Hint: Convert both mixed numbers to improper fractions before applying keep-change-flip.

Answer:

\frac{3}{2} \div \frac{15}{4} = \frac{3}{2} \times \frac{4}{15} = \frac{12}{30} = \frac{2}{5}

Related Terms

Fraction — Parent concept — the numbers being divided
Fraction Rules — Summary of all fraction operations
Numerator — Top number of a fraction
Denominator — Bottom number of a fraction
Improper Fraction — Often the result after dividing fractions
Mixed Number — Must be converted before dividing
Proper Fraction — A fraction less than one
Least Common Denominator — Needed for adding, not dividing, fractions
Decimal — Alternate form a fraction can be converted to
Ratio — Division relationship fractions can represent