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# Multiplying Decimals

If you already know how to add decimals, you will be able to easily multiply decimals. For example, say you're multiplying a decimal by a whole number, like $0.13×3$ .

This is the same as adding the decimal three times: $0.13+0.13+0.13$ . You can think of it as a simple word problem. If there are three friends who each have 13 cents, they have a total of 39 cents between them, or \$0.39.

## Multiplying a decimal by another decimal

It gets a bit trickier when you multiply a decimal by another decimal. Let's look at the problem: $0.12×0.9$ . Since the number 0.9 is less than 1, what does it mean to add up the first decimal number 0.9 times?

Example 1

This is where it comes in handy to remember that decimals are simply another way of writing fractions that have powers of 10 in the denominator. Multiplying a number by 0.9 is the same thing as finding nine-tenths of that number. You can easily rewrite $0.12×0.9$ as

$\frac{12}{100}×\frac{9}{10}$

When you multiply the numerators and denominators, you would end up with $\frac{108}{1000}$ . This fraction is the same as the decimal number 0.108.

Of course, there is an easier way to multiply decimals than to convert them to fractional notation every time.

## The standard algorithm for multiplying decimals

To start out, you simply multiply the two numbers as if they were whole numbers. Don't bother lining up the decimal points.

Next, you count the total number of places to the right of the decimal points in both numbers that you are multiplying. Make a note of this number "n". Start from the right and move n places to the left once you have your answer from multiplying. Place your decimal point at this location, and you'll have the correct answer.

Example 2

Let's practice this concept by multiplying $3.1×5.06$ .

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

$\begin{array}{cc}& \hfill 506\\ ×& \hfill \underset{_}{\phantom{\rule{10pt}{0ex}}31}\\ & \hfill 506\\ & \hfill \underset{_}{15180}\\ & \hfill 15686\end{array}$

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

$15.686$

You can check logically to see if this answer fits. Since 3.1 is close to 3 and 5.06 is close to 5, you would expect the answer to be close to 15. And it is!

Why does this work? Remember that what you're really doing is multiplying fractions. 3.1 actually means $\frac{31}{10}$ , and 5.06 means $\frac{506}{100}$ . When we multiply these fractions, we get $10×1000=1000$ 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 are really doing is multiplying powers of 10 in the denominators of the fractions.

## Multiplying decimals by 10, 100, and 1000

As we've already learned, multiplying decimals is the same as multiplying fractions with powers of 10 in the denominator. So there is a simple shortcut when we multiply decimals by 10, 100, 1000, or other powers of 10.

Take the problem $83.27×10=1000$ . Note that the digits in 83.27 and 832.7 are the same digits, but the decimal point has shifted one spot to the right in the answer. This is because, when you multiply a decimal by 10, you move the decimal point one place to the right because 10 has one 0 over 1.

The same concept is used for problems with decimals multiplied by other powers of 10.

Example 3

Multiply $34.826×100$

Because 100 has two zeros, you move the decimal point two places to the right to get the answer, which is

$3482.6$

Example 4

Multiply $83.47×1000$

Because 1000 has three zeros, you move the decimal point three places to the right to get the answer. Because there are not three digits to the right of the decimal point, you must add a zero to get the correct answer, which is

$83470$

## Get help learning about multiplying decimals

Multiplying decimals can be a little tricky when your student first learns it. Getting them help from a professional math tutor can give them a leg up in their math class and help them understand how to multiply decimals the right way. The 1-on-1 attention a tutor provides ensures your student has the chance to correct mistakes right away. To learn more about how tutoring can help your student learn to multiply decimals, contact the Educational Directors at Varsity Tutors today.

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