How to Divide Fractions
Dividing fractions uses the multiply-by-reciprocal method: flip the second fraction and multiply. This guide explains why this works and how to apply it confidently.
The Multiply-by-Reciprocal Method
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and denominator. This method works because dividing by a number is the same as multiplying by its reciprocal.
Step-by-Step Method
Write the Division Problem
Start with your division problem in the form a/b ÷ c/d
Example: 3/4 ÷ 2/5
Find the Reciprocal of the Second Fraction
Flip the second fraction (the one you're dividing by)
Example: Reciprocal of 2/5 is 5/2
Change Division to Multiplication
Replace the ÷ with × and use the reciprocal
Example: 3/4 ÷ 2/5 becomes 3/4 × 5/2
Multiply the Fractions
Multiply numerators together and denominators together
Example: (3 × 5)/(4 × 2) = 15/8
Simplify the Result
Simplify the fraction and convert to mixed number if needed
Example: 15/8 = 1 7/8
When You'd Use This
- Recipe Scaling: Dividing recipe quantities when serving fewer people
- Rate Calculations: Finding unit rates from fractional measurements
- Distribution Problems: Sharing fractional amounts among groups
- Engineering: Calculating ratios and proportions in technical work
Why This Method Works
The multiply-by-reciprocal method works because division is the inverse of multiplication. When you divide by a number, you're asking "how many times does this number fit into the dividend?"
Multiplying by the reciprocal gives the same result because the reciprocal "undoes" the original fraction. For example, multiplying by 1/2 then by 2/1 brings you back to where you started.
For a deeper mathematical explanation, see Wolfram MathWorld's article on fractions.
Frequently Asked Questions
Always flip the second fraction (the divisor). The first fraction stays as is.
Convert mixed numbers to improper fractions first, then use the standard method.
Yes, you can simplify by canceling common factors between any numerator and any denominator before multiplying, which often makes the calculation easier.