# Dividing Whole Numbers by Decimals

Steps for dividing whole number by decimal:

1. Simultaneously move decimal point to the right in both numbers, until you get whole number, instead of a decimal.
2. Now, you have two whole numbers. Divide them, using long division, except that you need to continue division, until you get zero remainder.

Why this works?

Because, we actually use equivalence of fractions here.

Suppose, you need to divide ${25}\div{0.08}$. This can be written as fraction $\frac{{25}}{{0.08}}$. We then multiply numerator and denominator by 10, which is equivalent to moving decimal point 1 position to the right: $\frac{{25}}{{0.08}}=\frac{{{25}\cdot{\color{red}{{{10}}}}}}{{{0.08}\cdot{\color{red}{{{10}}}}}}=\frac{{250}}{{0.8}}$.

We still didn't obtain whole number in denominator, so continue moving: $\frac{{{250}\cdot{\color{red}{{{10}}}}}}{{{0.8}\cdot{\color{red}{{{10}}}}}}=\frac{{2500}}{{8}}$.

Now, we can perform division.

Note, that you can speed up this process, by noting, that ${0.08}$ has two decimal places. This means, that you can right away multiply numerator and denominator by ${{10}}^{{2}}={100}$.

Also, when we divided whole numbers, we left non-zero remainder. That's because we didn't know about decimals. But now, you should continue dividing, until you get zero remainder.

Example 1. Calculate ${25}\div{0.08}$.

$\begin{array}{r}\color{blue}{3}\color{green}{1}\color{orange}{2}\\8\hspace{1pt})\overline{\hspace{1pt}2500}\\-\underline{\color{blue}{24}}\phantom{00}\\10\phantom{0}\\-\hspace{1pt}\underline{\phantom{1}\color{green}{8}}\phantom{0}\\20\\-\underline{\color{orange}{16}}\\4\\\end{array}$

Now, there is non-zero remainder, but we need to continue. For this, add decimal and required number of zeros after it.

$\begin{array}{r}\color{blue}{3}\color{green}{1}\color{orange}{2}\huge{\color{red}{.}}\large{\color{magenta}{5}}\\8\hspace{1pt})\overline{\hspace{1pt}2500\huge{\color{red}{.}}\large{\color{brown}{0}}}\\-\underline{\color{blue}{24}}\phantom{00}\phantom{\huge{.}\large{0}}\\10\phantom{0}\phantom{\huge{.}\large{0}}\\-\hspace{1pt}\underline{\phantom{1}\color{green}{8}}\phantom{0}\phantom{\huge{.}\large{0}}\\20\phantom{\huge{.}\large{0}}\\-\underline{\color{orange}{16}}\phantom{\huge{.}\large{0}}\\4\phantom{\huge{.}}\large{0}\\-\underline{\color{magenta}{4\phantom{\huge{.}}\large{0}}}\\\color{cyan}{0}\\\end{array}$

Note, that we've added one zero in order to complete division, but in general, you, probably, need to add more than one zero.

So, ${\color{purple}{{{25}\div{0.08}={312.5}}}}$.

There can be a situation, when you've prepared numbers for division, but dividend is less than divisor.

That's not a problem. Move decimal point in the dividend only, until you get the number, greater than divisor. Perform division and then undo moving, i.e. move decimal point to the left.

Example 2. Calculate ${8}\div{12.5}$.

First, simultaneously move decimal point one position to the right: ${80}\div{125}$.

Dividend is less than divisor, so move decimal point to the right: ${80.0}{\color{blue}{{\to}}}{800.0}$ (we moved point 1 time).

Now we can divide 800 by 125.

And now, don't forget, that we moved decimal point in dividend 1 position to the right, so need to undo it.

Move decimal point 1 position to the left: ${6.4}{\color{blue}{{\to}}}{0.64}$.

So, ${\color{purple}{{{8}\div{12.5}={0.64}}}}$.

Now, it is your turn. Take pen and paper and solve following problems:

Exercise 1. Find ${8}\div{0.25}$.

Exercise 2. Calculate ${48}\div{38.4}$.

Exercise 3. Find $-{2}\div{6.4}$.
Exercise 4. Find $-{48}\div{\left(-{0.3}\right)}$.
Exercise 5. Find ${2}\div{5.12}$.