# Multiplying and Dividing with Decimals

## Multiplying Decimals

Suppose you're multiplying a decimal by a whole number, say $0.12×3$ .

This is the same as adding the decimal three times: $0.12+0.12+0.12$ . You can think of it as follows: If three friends each have $12$ cents, together, they have a total of $36$ cents.

It's a bit trickier when both numbers are decimals. Take the problem $0.12×0.9$ . The number $0.9$ is less than $1$ , so what does it mean to add up the first decimal $0.9$ times?

Remember that decimals are just another way of writing fractions that have powers of $10$ in the denominator. Multiplying a number by $0.9$ is the same as finding nine-tenths of that number. So you could rewrite the problem $0.12×0.9$ as

$\frac{12}{100}×\frac{9}{10}$ .

Then you would multiply numerators and denominators to get $\frac{108}{1000}$ . This fraction is the same as the decimal $0.108$ .

Of course, you don't have to convert to fraction notation every time.

Standard Algorithm for Multiplying Decimals

First just multiply the numbers as if they were whole numbers . (Don't line up the decimal points!)

Then count the total number of places to the right of the decimal point in BOTH numbers you're multiplying. Let's call this number $n$ . In your answer, start from the right and move $n$ places to the left, and put a decimal point.

Example:

Multiply $3.1×5.06$ .

Step 1: Multiply the numbers, ignoring the decimal point.

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\underset{_}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\text{\hspace{0.17em}}1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\\ \text{\hspace{0.17em}}\underset{_}{+\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}8\text{\hspace{0.17em}}\text{\hspace{0.17em}}0}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\text{\hspace{0.17em}}\text{\hspace{0.17em}}8\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\end{array}$

Step 2: In $3.1$ , there is $1$ place to the right of the decimal point. In $5.06$ , there are $2$ . So, since $1+2=3$ , move in $3$ decimal places from the right in your answer.

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You can check that this is reasonable. $3.1$ is close to $3$ , and $5.06$ is close to $5$ , so we expect an answer close to $15$ . And we got one!

Why does this work? Again, what you're really doing is multiplying fractions. $3.1$ means $\frac{31}{10}$ , and $5.06$ means $\frac{506}{100}$ . When we multiply these fractions, we get $10×100=1000$ in the denominator, so the final answer is expressed in thousandths. When you add the total number of places to the right of the decimal points in the factors, what you're really doing is multiplying powers of ten in the denominators of the fractions.

## Dividing with Decimals

Dividing with decimals is a bit more 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}}\text{\hspace{0.17em}}\underset{_}{3\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}}\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}}\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}}\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}}\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}}\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}}\text{\hspace{0.17em}}\underset{_}{3\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}}\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}}\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}}\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}}\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}}\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!