Multiplying / Dividing Fractions
Multiplying and dividing fractions are much simpler processes than adding and subtracting.
Warning - Doing Too Much
A common mistake that I see is finding a common denominator for the two fractions like we did when we were doing addition and subtraction. There's no harm in making the two fractions you're given into equivalent ones with a common denominator - it just isn't necessary. Also, doing that adds several additional steps which means more opportunities to make an error somewhere along the way. Ultimately, you're better off not bothering since it isn't a necessary step.
To multiply two fractions:
- Multiply the numerators together to get the numerator of the result.
- Multiply the denominators together to get the denominator of the result.
- Reduce the fraction.
That's really all there is to it: Multiply the tops, multiply the bottoms, reduce and you're finished. Division is almost very similar but with one extra step:
- Flip the second fraction over, e.g. 2/3 would become 3/2, 1/3 would become 3/1, etc.
- Change the division symbol to multiplication and follow the steps from the previous procedure.
Example 1
Multiply (2/3) · (5/8).
The numerator of our answer will be 2 · 5 = 10.
The denominator will be 3 · 8 = 24.
This makes our answer, 10 / 24, but that can be reduced by dividing the numerator and denominator by 2 to give us 5 / 12 as our final answer.
We could summarize the whole process with this equation:
Example 3
Evaluate (3/5) ÷ (25/3).
At first glance this looks very similar to Example 2 but remember that division has an extra step. You start by converting it to an "equivalent" multiplication problem by flipping over the second fraction. In this case, that means changing 25/3 to 3/25:
(3/5) ÷ (25/3) = (3/5) · (3/25)
Now we can just follow the steps for multiplying two fractions. The numerator of our answer will be 3 · 3 = 9.
The denominator will be 5 · 25 = 125.
This makes our answer 9 / 125 which can't be reduced any further.
We could summarize the whole process with this equation:
Example 2
Multiply (3/5) · (25/3).
The numerator of our answer will be 3 · 25 = 75.
The denominator will be 5 · 3 = 15.
This makes our answer, 75 / 15, but that can be reduced by dividing the numerator and denominator by 15 to give us 5 / 1 = 5 as our final answer.
We could summarize the whole process with this equation:
Example 4
Evaluate (5/8) ÷ (19/4).
We start off by converting this to a multiplication problem by flipping over the second fraction:
(5/8) ÷ (19/4) = (5/8) · (4/19)
Now we'll follow the steps for multiplying two fractions. The numerator of our answer will be 5 · 4 = 20.
The denominator will be 8 · 19 = 152.
This makes our answer, 20 / 152, but that can be reduced by dividing the numerator and denominator by 4 to give us 5 / 38 as our final answer.
We could summarize the whole process with this equation:
