How to Divide Fractions

Mathematicians didn’t want to muck up fraction division by making you do something as complicated as actually dividing, so they devised a way to use multiplication instead. To divide one fraction by another fraction, change the problem to multiplication:

1. Change the division sign to a multiplication sign.

2. Change the second fraction to its reciprocal.

   Switch around the numerator (top number) and denominator (bottom number).

3. Solve the problem using fraction multiplication.

When dividing fractions, you may have to reduce your answer or change it from an improper fraction to a mixed number.

Sample questions

1. Divide 5/8 by 3/7.

   

   Change the division to multiplication:

   

   Solve the problem using fraction multiplication:

   

   The answer is an improper fraction (because the numerator is greater than the denominator), so change it to a mixed number. Divide the numerator by the denominator and put the remainder over the denominator:

   

2. Solve

   

   Change the division to multiplication:

   

   Notice that you have a 5 in one of the numerators and a 10 in the other fraction’s denominator, so you can cancel out the common factor, which is 5; that would change your problem to

   

   Or you can simply do your calculations and reduce the fraction later, as I do here. Solve by multiplying these two fractions:

   

   This time, the numerator and denominator are both divisible by 5, so you can reduce them:

   

   Because the numerator is greater than the denominator, the fraction is improper, so change it to a mixed number:

   1-3/4

Practice questions

1. Divide 1/4 by 6/7.

2. Find

   

3. Divide 8/9 by 3/10.

4. Solve

   

Following are answers to the practice questions:

1. 

   First, change the problem to multiplication, multiplying by the reciprocal of the second fraction:

   

   Now complete the problem using fraction multiplication:

   

2. 

   Change the problem to multiplication, using the reciprocal of the second fraction:

   

   Complete the problem using fraction multiplication:

   

   Both the numerator and denominator are divisible by 5, so reduce the fraction by this factor:

   

   They’re still both divisible by 3, so reduce the fraction by this factor:

   

3. 

   Change the problem to multiplication, using the reciprocal of the second fraction:

   

   Complete the problem using fraction multiplication:

   

   The numerator is greater than the denominator, so change this improper fraction to a mixed number:

   

4. 

   Change the problem to multiplication, using the reciprocal of the second fraction:

   

   Complete the problem using fraction multiplication:

   

   The numerator is greater than the denominator, so change this improper fraction to a mixed number:

   

   Now the numerator and the denominator are both divisible by 3, so reduce the fractional part of this mixed number by a factor of 3:

   

   The numerator and denominator are now both divisible by 7, so reduce the fractional part by this factor: