So far I have avoided division of fractions. This is not hard, in fact, it’s almost as easy as multiplying but I would like to show why the method works.
First consider 6 ÷ 2. This problem is asking the question, “How many 2’s fit into 6?”. The answer is 3 sets of 2’s make up 6. If the problem was 6 ÷ 3, the question would be “How many 3’s fit into 6?”. The answer here is there are 2 sets of 3’s that make up 6. Of course, the answer is not always an integer, Sometimes there are leftover numbers. For example, how many 2’s can make up 7, in other words, 7 ÷ 2. The answer is there are 3 sets of 2’s that can fit into 7 but there will be 1 left over as sets of 2’s do not exactly make up 7. Now remember that fractions are indicating division as well so the above problems are equivalent to
\[\frac{6}{2}{,}\hspace{0.33em}\hspace{0.33em}\frac{6}{3}{,}\hspace{0.33em}\hspace{0.33em}\frac{7}{2}
\]
So what would \[
{1}\hspace{0.33em}\div\hspace{0.33em}\frac{1}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{\frac{1}{2}}
\]
mean? It means the same as before: how many one-halves fit into 1? Now the answer is 2 because there are two halves in a whole. What about \[
{2}\hspace{0.33em}\div\hspace{0.33em}\frac{1}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{\frac{1}{2}}
\]? Can you see that the answer is 4? Because if there are two things split into halves, then there will be 4 halves.
Let’s now look at \[
{3}\hspace{0.33em}\div\hspace{0.33em}\frac{3}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{\frac{3}{4}}
\]
Looks like there are 4 three-quarters that fit into 3, and that is correct. Let’s look at one more. What about \[
\frac{3}{4}\hspace{0.33em}\div\hspace{0.33em}\frac{1}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{\frac{3}{4}}{\frac{1}{4}}
\]. Can you see that there are 3 one-quarters that fit into three-quarters?
All of the above answers can be obtained by multiplying and remembering that integers like 6 can also be shown as a fraction: \[
\frac{6}{1}
\] as 6 divided by 1 is still 6.
So now let’s look at the previous problems. You can convert a fraction division problem by inverting the fraction in the denominator then multiplying:
\[{1}\hspace{0.33em}\div\hspace{0.33em}\frac{1}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{\frac{1}{2}}\hspace{0.33em}{=}\hspace{0.33em}\frac{1}{1}\hspace{0.33em}\times\hspace{0.33em}\frac{2}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{1}\hspace{0.33em}{=}\hspace{0.33em}{2}
\]
The inverted fraction is called a reciprocal. A simply way to remember how to do fraction division is the phrase: invert and multiply.
Now let’s do the other ones:
\[{3}\hspace{0.33em}\div\hspace{0.33em}\frac{3}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{\frac{3}{4}}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{3}}{1}\hspace{0.33em}\times\hspace{0.33em}\frac{4}{\rlap{/}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{4}{1}\hspace{0.33em}{=}\hspace{0.33em}{4}
\]
And finally
\[\frac{3}{4}\hspace{0.33em}\div\hspace{0.33em}\frac{1}{4}\hspace{0.33em}{=}\hspace{0.33em}\frac{\frac{3}{4}}{\frac{1}{4}}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{\rlap{/}{4}}\hspace{0.33em}\times\hspace{0.33em}\frac{\rlap{/}{4}}{1}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{1}\hspace{0.33em}{=}\hspace{0.33em}{3}
\]
I will do more examples in my next post including division with mixed numbers.