8.10 Divide whole numbers by unit fractions

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Do you remember how a fraction can represent division?

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When we divide by a whole number, such as $12\div4$12÷​4, we ask the question "how many groups of $4$4 fit into $12$12?" It's just like thinking about "what number fills in the blank: $4\times\editable{}=12$4×=12".

In this case, there are $3$3 whole groups of $4$4 in $12$12, so the result is $3$3.

We can think about dividing by a unit fraction in a similar way. The division $2\div\frac{1}{3}$2÷​13​ is equivalent to asking the question "how many parts of size $\frac{1}{3}$13​ fit into $2$2 wholes?"

If we split two wholes up into thirds, we can see that there are $3$3 thirds in each whole, and so there are $2\times3=6$2×3=6 thirds in total.

The same thing happens for dividing by other unit fractions. If we calculated $3\div\frac{1}{5}$3÷​15​ this time, each of the three wholes will be divided into $5$5 fifths:

So $3\div\frac{1}{5}$3÷​15​ is the same as $3\times5=15$3×5=15.

Notice that this is just like thinking about "what number fills in the blank: $\frac{1}{5}\times\editable{}=3$15​×=3". We know that $\frac{1}{5}\times15=3$15​×15=3, so it makes sense that $3\div\frac{1}{5}=15$3÷​15​=15.

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