Dividing Decimals

Dividing with decimals is a bit difficult. These days, most teachers don't mind much if you use a calculator. But it's good to know how to do it yourself, too, and you always need to be good at estimating the answer, so you can make sure the calculator's answer is reasonable.

Recall that in the problem $x÷y=z$ , also written

$\begin{array}{c}\\ y\end{array}\begin{array}{c}\hfill z\\ \hfill \overline{)x}\end{array}$

$x$ is called the dividend , $y$ is the divisor , and $z$ is the quotient .

Step 1: Estimate the answer by rounding . You'll use this estimate to check your answer later.

Step 2: If the divisor is not a whole number, then move the decimal place $n$ places to the right to make it a whole number. Then move the decimal place in the dividend the same number of places to the right (adding some extra zeros if necessary.)

Step 3: Divide as usual. If the divisor doesn't go in evenly, add zeros to the right of the dividend and keep dividing until you get a $0$ remainder, or until a repeating pattern shows up.

Step 4: Put the decimal point in the quotient directly above where the decimal point now is in the dividend.

Example:

Divide.

$0.45÷3.6$

Step 1: Since the divisor is greater than the dividend, we will get an answer less than $1$ . Since $0.45$ is about one tenth as big as $3.6$ , we expect an answer close to $0.1$ .

Step 2: The divisor is not a whole number, so move the decimal point one place to the right to make it a whole number. Move the decimal point in the dividend one place to the right also.

$36\overline{)4.5}$

Step 3: Divide normally, adding extra zeros to the right of $4.5$ when you run out.

$\begin{array}{l}\begin{array}{c}\\ 36\end{array}\begin{array}{c}\hfill 125\\ \hfill \overline{)4.500}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{3\text{\hspace{0.17em}}\text{\hspace{0.17em}}6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}90\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{72}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}180\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{180}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}$

Step 4: Put the decimal point in the quotient directly above the decimal point in the dividend.

$\begin{array}{l}\begin{array}{c}\\ 36\end{array}\begin{array}{c}\hfill 0.125\\ \hfill \overline{)4.500}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{3\text{\hspace{0.17em}}\text{\hspace{0.17em}}6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}90\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{72}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}180\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{_}{180}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}$

We get $0.125$ .

Step 5: Compare with your initial estimate. $0.125$ is close to $0.1$ , so we're good!

Sometimes, it's easier to use mental math to solve a decimal division problem. This is a good strategy when you can see that if you move the decimal points around, you can change the problem into one you've memorized the answer to.

Example:

Divide.

$0.42÷70$

We know that $42÷7=6$ .

If the dividend is decreased by a factor of $10$ , then the quotient will also decrease by a factor of $10$ .

$\begin{array}{l}42÷7=6\\ 4.2÷7=0.6\\ 0.42÷7=0.06\end{array}$

And if the divisor is increased by a factor of $10$ , then the quotient will decrease by a factor of $10$ .

$0.42÷70=0.006$

So, the answer is $0.006$ .