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How to Divide Fractions
The idea of dividing a fraction by another fraction may sound daunting but there's a simple process we can follow to solve these equations. We know from How to Multiply Fractions, that multiplying fractions is quite a straight-forward process so we're going to use that again now.
Multiplying and dividing are inverse functions so to 'turn' our division equation into multiplication, we need to use the inverse or rather 'reciprocal' of one of our fractions.
Check out this example:
Step 1: Change the division equation to a multiplication equation by finding the reciprocal of the second fraction
Step 2: Solve by multiplying the numerators and multiplying the denominators
Step 3: Simplify (if needed)
If you start with a mixed number, first convert it to an improper fraction then solve and simplify:
If you need to divide a whole number by a fraction, write the whole number as a fraction with 1 as the denominator then solve and simplify:
But what are we actually calculating? When we divide a fraction by another fraction, we’re finding out how many groups of that fraction go into the other. In the equation 1/2 ÷ 1/4, what we're trying to calculate is How many quarters are in one half? Check out this visual representation:
Picture one half. Now we've got to find out how many quarters 'fit' into that half. In this case, it is very easy to see on our diagram that two quarters 'fit' onto one half.
In the details to the right of the diagram, we can see how you could solve the equation by using the inverse operation (multiplication) and the reciprocal of 1/4, which is 4/1. Our answer of 4/2 can be simplified so we end up with 2.
That means 1/2 ÷ 1/4 = 2
Let's look at another example. If we were calculating 3/5 ÷ 1/3, we would solve it by finding the reciprocal of 1/3 which is 3/1 and multiplying that by 3/5.
3/5 x 3/1 = (3x3)/(5x1)
= 1 4/5
Go back to visuals if you need. What the answer of one and four-fifths tells us is that 1/3 'fits' onto 3/5 (picture that now) but a second 1/3 doesn't quite. Only 4/5 of another third would fit. Hence our answer is not a 'tidy' number like in the previous equation. But it can still be done.
Last example: 5/3 ÷ 2/7
5/3 x 7/2 = (5x7)/(3x2)
= 5 5/6
Five and five-sixths of the fraction 2/7 'fit' into the fraction 5/3.
Have a go at solving some equations yourself now. Even if this method seems a little weird to start out, trust that it'll become second-nature in no time!
If you're after more fractions guides, check these out: