## DIVISION

Consider $0.08 \div 0.005$.

To do this problem I could try drawing a picture of $0.08$ in a $1 \leftarrow 10$ machine and attempt to make sense of finding groups of $0.005$ in that picture, that is groups of five dots three places to the right of where they should be. Although possible to do, it seems hard to keep straight in my head and it makes my brain hurt!

Here’s a piece of advice from a mathematician: **Avoid hard work!** Mathematicians will, in fact, work very hard to avoid hard work!

Since converting decimals to fractions makes the multiplication of fractions easier, let’s do the same thing for division.

Here are some examples.

**Example**: Examine $0.08 \div 0.005$.

A fraction is a number that is an answer to a division problem. And, conversely, we can think of a division problem as a fraction. For example, the quantity $0.08 \div 0.005$ is really this “fraction” $\dfrac {0.08}{0.005}$.

(It is okay to have non-whole numbers as numerators and denominators of fractions.)

To make this fraction look friendlier, let’s multiply the top and bottom by factors of ten.

$\dfrac {0.08 \times 10 \times 10 \times 10}{0.005 \times 10 \times 10 \times 10} = \dfrac{80}{5}$

The division problem $\dfrac{80}{5}$ is much friendlier. It has the answer $16$.

(Use a $1 \leftarrow 10$ machine to compute it if you like!)

**Example**: Examine $\dfrac{8.5}{100}$.

Dividing by $100$ is the same as multiplying by $\dfrac{1}{100}$. So, $\dfrac{8.5}{100}$, for example, can be thought of as

$\dfrac{1}{100} \times 8.5 = \dfrac{1}{100} \left(8+\dfrac{5}{10}\right)=\dfrac{8}{100} + \dfrac{5}{1000} = 0.085$

**Example**: Examine $1.51 \div 0.07$.

If I was asked to compute $1.51 \div 0.07$, I would personally do this problem instead:

$\dfrac{1.51 \times 10 \times 10}{0.07 \times 10 \times 10}= \dfrac{151}{7}$.

I see the answer $21 \dfrac{4}{7}$. If that same person asked me to give the answer as a decimal, then I would have to convert the fraction $\dfrac{4}{7}$ into a decimal. And that is possible. The next lesson explains how!