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Decimal Division (decimal divisors)

When dividing by a decimal, the easiest way to deal with it is to get the decimal out of the divisor.  In math, the rule is, you can do anything to a number, as long as you do the same to every other number in the problem or equation.  So, you multiply the divisor by 10 (or 100 or 1000) or whatever you need in order to push the decimal all the way to the right.  Then you multiply the dividend by the same amount to keep the problem balanced.  

To represent this process, we usually just move the decimal point.  If you have a decimal point in your divisor, move it all the way to the right of the number (so the divisor becomes a whole number).  Then move the decimal point in the dividend the same number of places to the right as you moved the decimal in the divisor (this maintains the balance in the problem).  Then you do normal decimal division: bring the decimal from the dividend up into the answer row and do the long division process. 

For more on the long division process, see lessons Long Division (1-digit divisor) and Long Division (multi-digit divisor) and for more on division with decimals see lesson Decimal Division (whole number divisors). Once students have master long division, they can work with decimals easily.

Bottom line, if there is a decimal in the divisor, move it all the way to the right and move the decimal in the dividend the same number of places to the right. 

Example: $7.85\div .05=$

Set the problem up:

$$\eqalign{.05&\overline{)7.85}\qquad&&\text{Write the problem in a house}}$$

Deal with the decimal in your divisor.  Move it all the way to the right. Count the number of place values you move it 

$$\eqalign{\underset{2 \text{ places}}{\underrightarrow{.05.}}&\overline{)7.85}\qquad&&\text{Move the decimal in the divisor all the way to the right.}}$$

$$\eqalign{05.&\overline{)7\underset{2 \text{ places}}{\underrightarrow{.85.}}}\qquad&&\text{Move the decimal in the dividend the same number of places to the right.}}$$

Now there is no decimal in the divisor.  To deal with the decimal in your dividend, bring it straight up, from the dividend to the answer row.  Then, proceed with long division, ignoring the decimal (but making sure to put answer digits above the proper dividend digits so that the decimal ends up in the right place).

$$\eqalign{\quad & \quad \quad\color{red}{ .} \\5&\overline{)785.0}\qquad&&\text{Bring the decimal up from the dividend into the answer row}}$$

Do the first "division" step.  How many times does 5 go into the first digit in the dividend?

$$\eqalign{ \quad & \; \color{red}{1}\quad.\qquad && \color{red}{\text{5 goes into 7 one time, write 1 over the 7}}\\5 & \overline{)785.0}\qquad && \quad \\&\!\! \underline{-5}  \qquad && \text{5 times 1 is five; subtract from 7}\\  &\;\; 2  \qquad && \text{7 minus 5 equals 2}}$$

Now, you bring down and divide into the new number you created when you brought down.

$$\eqalign{ \quad & \; \color{red}{1}\color{green}5\;\; .&& \color{green}{\text{5 goes into 28 five times, write the 5 over the 8}}\\5 & \overline{)785.0}\qquad && \quad \\&\!\! \underline{-5} \color{green}{\downarrow} \qquad && \\  &\;\; 2\color{green}{8}  \qquad && \text{Bring down the 8 and put it next to the 2}\\&\!\!\underline{-25}\qquad &&\text{5 times 5 is 25}\\&\;\;\;3\qquad&&\text{28 minus 25 is 3}}$$

There is one more digit in the dividend, so you'll bring down one more time, then divide again into the new number.

$$\eqalign{ \quad & \; \color{red}{1}\color{green}5\color{blue}{7}.\qquad && \color{blue}{\text{5 goes into 35 seven times, write the 7 over the 5}}\\5 & \overline{)785.0}\qquad && \quad \\&\!\! \underline{-5} \color{green}{\downarrow} \qquad && \\  &\; 2\color{green}{8}  \qquad && \\&\!\!\!\underline{-25}\color{blue}{\downarrow}\qquad &&\\&\;\;3\color{blue}{5}\qquad&&\text{Bring down the 5}\\&\!\underline{-35}\qquad&&\text{7 times 5 is 35}\\&\quad1\qquad&&\text{35 minus 35 is 0, you are done}}$$

The answer is now at the top of the "house."  $\color{purple}{7.85 \div.05 = 157 }$ 

Overall, the process of dividing with a decimal in the dividend is just like normal long division, just bring the decimal up into the answer (and, ideally, do it before you begin, so you don't forget!).

Note: Remember, if there is no decimal in the dividend, then the dividend is a whole number and the decimal is invisible but just to the right of the last digit.   You can move it to the right more places, just add a placeholder zero for each place you move it. 

Practice Problems: