Learning Objective(s)

·Multiply two or more decimals.

·Multiply a decimal by a power of 10.

·Divide by a decimal.

·Divide a decimal by a power of 10.

·Solve application problems that require decimal multiplication or division.

As with whole numbers, sometimes you run into situations where you need to multiply or divide decimals. And just as there is a correct way to multiply and divide whole numbers, so, too, there is a correct way to multiply and divide decimals.

Imagine that a couple eats dinner at a Japanese steakhouse. The bill for the meal is $58.32--which includes a tax of $4.64. To calculate the tip, they can double the tax. So if they know how to multiply $4.64 by 2, the couple can figure out how much they should leave for the tip.

Here's another problem. Andy just sold his van that averaged 20 miles per gallon of gasoline. He bought a new pickup truck and took it on a trip of 614.25 miles. He used 31.5 gallons of gas to make it that far. Did Andy get better gas mileage with the new truck?

Both of these problems can be solved by multiplying or dividing decimals. Here's how to do it.

Multiplying decimals is the same as multiplying whole numbers except for the placement of the decimal point in the answer. When you multiply decimals, the decimal point is placed in the product so that the number of decimal places in the product is the sum of the decimal places in the factors.

Let's compare two multiplication problems that look similar: 214 · 36, and 21.4 · 3.6.

Notice how the digits in the two solutions are exactly the same - the multiplication does not change at all. The difference lies in the placement of the decimal point in the final answers: 214 · 36 = 7,704, and 21.4 · 3.6 = 77.04.

To find out where to put the decimal point in a decimal multiplication problem, count the total number of decimal places in each of the factors.

21.4     the first factor has one decimal place

  3.6     the second factor has one decimal place

          77.04     the product will have 1 + 1 = 2 decimal places

Note that the decimal points do not have to be aligned as for addition and subtraction.

Sometimes you may need to insert zeros in front of the product so that you have the right number of decimal places. See the final answer in the example below:

If one or more zeros occur on the right in the product, they are not dropped until after the decimal point is inserted.

Take a moment to multiply 4.469 by 10. Now do 4.469 · 100. Finally, do 4.469 · 1,000. Notice any patterns in your products?

Notice that the products keep getting greater by one place value as the multiplier (10, 100, and 1,000) increases. In fact, the decimal point moves to the right by the same number of zeros in the power of ten multiplier.

You can use this observation to help you quickly multiply any decimal by a power of ten (10, 100, 1,000, etc).

To divide decimals, you will once again apply the methods you use for dividing whole numbers. Look at the two problems below. How are the methods similar?

Notice that the division occurs in the same way--the only difference is the placement of the decimal point in the quotient.

But what about a case where you are dividing by a decimal, as in the problem below?

In cases like this, you can use powers of 10 to help create an easier problem to solve. In this case, you can multiply the divisor, 0.3, by 10 to move the decimal point 1 place to the right. If you multiply the divisor by 10, then you also have to multiply the dividend by 10 to keep the quotient the same. The new problem, with its solution, is shown below.

Often, the dividend will still be a decimal after multiplying by a power of 10. In this case, the placement of the decimal point must align with the decimal point in the dividend.

Recall that when you multiply a decimal by a power of ten (10, 100, 1,000, etc), the placement of the decimal point in the product will move to the right according to the number of zeros in the power of ten. For instance, 4.12 · 10 = 41.2.

Multiplication and division are inverse operations, so you can expect that if you divide a decimal by a power of ten, the decimal point in the quotient will also correspond to the number of zeros in the power of ten. The difference is that the decimal point moves to the right when you multiply; it moves to the left when you divide.

In the examples above, notice that each quotient still contains the digits 4469--but as another 0 is added to the end of each power of ten in the divisor, the decimal point moves an additional place to the left in the quotient.

Now let's return to the two problems from the beginning of this section. You know how to multiply and divide with decimals now. Let's put that knowledge to the test.

Learning to multiply and divide with decimals is an important skill. In both cases, you work with the decimals as you have worked with whole numbers, but you have to figure out where the decimal point goes. When multiplying decimals, the number of decimal places in the product is the sum of the decimal places in the factors. When dividing by decimals, move the decimal point in the dividend the same number of places to the right as you move the decimal point in the divisor. Then place the decimal point in the quotient above the decimal point in the dividend.

This reading is taken from the Developmental Math Open Program created by The NROC Project. It is available under a Creative Commons license.