Formula
To divide fractions: keep the first fraction, change ÷ to ×, flip the second fraction (take its reciprocal), then multiply across and simplify.
How to use this calculator
Enter the numerator and denominator of each fraction. The calculator divides the first by the second, showing the reciprocal step, simplified result, mixed number (where applicable), decimal, and percentage.
Dividing fractions formula
Dividing by a fraction is identical to multiplying by its reciprocal:
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$$
This works because division and multiplication are inverse operations. Multiplying by d/c "undoes" the division by c/d. The result is then simplified by dividing both parts by their GCD.
What is a reciprocal?
The reciprocal of a fraction is formed by swapping its numerator and denominator:
$$\text{Reciprocal of } \frac{c}{d} = \frac{d}{c}$$
A fraction and its reciprocal always multiply to 1: (a/b) × (b/a) = ab/ab = 1. This is why replacing division with multiplication by the reciprocal preserves the correct answer. The reciprocal is sometimes called the multiplicative inverse.
Special cases:
- Reciprocal of a whole number n is 1/n (e.g. reciprocal of 4 is 1/4)
- Reciprocal of 1 is 1
- Zero has no reciprocal — division by zero is undefined
Worked examples
Example 1: standard division
Divide 3/4 by 2/5. Flip 2/5 to get 5/2, then multiply:
$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$
Numerators: 3 × 5 = 15. Denominators: 4 × 2 = 8. GCD(15, 8) = 1, already simplified. As a mixed number: 1 7/8.
Example 2: result simplifies to a whole number
Divide 2/3 by 4/6. Flip 4/6 to get 6/4, then multiply:
$$\frac{2}{3} \div \frac{4}{6} = \frac{2}{3} \times \frac{6}{4} = \frac{12}{12} = 1$$
The result 12/12 simplifies to 1. This makes sense: 2/3 and 4/6 are equivalent fractions, so their quotient is 1.
Example 3: dividing a fraction by itself
Any non-zero fraction divided by itself equals 1:
$$\frac{5}{6} \div \frac{5}{6} = \frac{5}{6} \times \frac{6}{5} = \frac{30}{30} = 1$$
| Problem | Reciprocal used | Raw product | Simplified |
|---|---|---|---|
| 1/2 ÷ 1/4 | 4/1 | 4/2 | 2 |
| 2/3 ÷ 1/2 | 2/1 | 4/3 | 1 1/3 |
| 3/4 ÷ 3/8 | 8/3 | 24/12 | 2 |
| 5/6 ÷ 2/3 | 3/2 | 15/12 | 5/4 = 1 1/4 |
| 7/8 ÷ 7/4 | 4/7 | 28/56 | 1/2 |
| 4/5 ÷ 8/15 | 15/8 | 60/40 | 3/2 = 1 1/2 |
Dividing by a whole number (and dividing a whole number by a fraction)
Write the whole number as a fraction over 1, then apply keep-change-flip:
$$\frac{3}{4} \div 3 = \frac{3}{4} \div \frac{3}{1} = \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4}$$
Dividing a fraction by n is the same as multiplying the denominator by n. This halves, thirds, or further reduces the fraction depending on n.
Dividing a whole number by a fraction inverts the relationship — the result is often larger than the original:
$$3 \div \frac{3}{4} = \frac{3}{1} \div \frac{3}{4} = \frac{3}{1} \times \frac{4}{3} = \frac{12}{3} = 4$$
"How many ¾ fit into 3?" — four times. Dividing by a fraction less than 1 always produces a result greater than the dividend.
Dividing mixed numbers
Convert each mixed number to an improper fraction before applying keep-change-flip:
$$2\frac{1}{2} \div 1\frac{1}{4} = \frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2$$
2½ → 5/2 (2 × 2 + 1 = 5). 1¼ → 5/4 (1 × 4 + 1 = 5). Flip 5/4 to get 4/5. Multiply: 5/2 × 4/5 = 20/10 = 2.
Negative fractions
The sign rules for dividing fractions are identical to those for multiplication:
- Positive ÷ positive = positive
- Negative ÷ negative = positive
- Positive ÷ negative = negative
- Negative ÷ positive = negative
$$\frac{-3}{4} \div \frac{1}{2} = \frac{-3}{4} \times \frac{2}{1} = \frac{-6}{4} = \frac{-3}{2}$$
The sign is determined before or after flipping — it makes no difference, since flipping the reciprocal preserves the sign of the second fraction's numerator and denominator independently.
Common mistakes
Flipping the first fraction instead of the second
Only the second fraction (the divisor) is flipped. 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8, not 4/3 × 2/5 = 8/15. The first fraction stays as written.
Flipping both fractions
Flipping both fractions changes the meaning of the calculation entirely. Only the divisor is inverted.
Finding a common denominator first
A common denominator is not needed for division. It is only required for addition and subtraction. Introducing a common denominator step here adds unnecessary work and creates opportunities for error.
Not converting mixed numbers first
Applying keep-change-flip to a mixed number without converting it to an improper fraction first produces a wrong answer. Always convert: 2½ = 5/2, not "2 and ½."
Frequently asked questions
How do you divide fractions?
Keep the first fraction, change ÷ to ×, flip the second fraction (take its reciprocal), multiply across, and simplify.
What is the reciprocal of a fraction?
The reciprocal of a/b is b/a — numerator and denominator are swapped. A fraction times its reciprocal always equals 1.
Do you need a common denominator to divide fractions?
No. Fraction division converts to multiplication via the reciprocal, and multiplication needs no common denominator.
How do you divide a fraction by a whole number?
Write the whole number as n/1 and apply keep-change-flip: a/b ÷ n = a/b × 1/n = a/(b×n).
How do you divide mixed numbers?
Convert each mixed number to an improper fraction, then apply keep-change-flip to the two improper fractions.
Can you divide a fraction by itself?
Yes — the result is always 1. Any non-zero number divided by itself equals 1.