Math Background

Multiplying and Dividing Decimals: Overview

In the next seven years, before your students graduate from high school, how many times will these students use a computer, work with a calculator, solve problems involving money, or use medical and scientific information? Multiplying and dividing decimals will often be involved with all of these activities. It is important that students have a solid understanding of these operations.

Candace measured rainfall in her rain gauge for one week. Her average daily measurement was 0.3 inches. How much rain fell during the week? This problem, which involves finding the product of a whole number and a decimal, can be solved in different ways. Students are familiar with multiplication of whole numbers and fractions.

You can write the factors as fractions.

7 x 0.3 = seven over one x three-tenths = twenty-one-tenths = 2one-tenth, or 2.1

For many decimal problems it is easier to multiply first and then to place the decimal point.

Multiply the factors as if they were whole numbers. Disregard the decimal point until you place it in the product. The rule for deciding how to place the decimal point in the product works for all decimal multiplication: The number of decimal places in the product must equal the total number of decimal places in the factors.

multiplication problem

This rule must be followed when multiplying two factors containing decimals. When José visited Toronto, Canada, on the school trip, the exchange rate for money was 1.52 Canadian dollars for 1 American dollar. If José had $7.25 to exchange, how much Canadian money would he receive?

multiplication problem

The total number of decimal places in the factors is 4. The product must have 4 decimal places. José would receive $11.02 of Canadian money.

Sometimes you need to write one or more zeros in the product before you can place the decimal point.

multiplication problem

You can write as many zeros as needed to the left of the product to place the decimal point.

An excellent way to determine if your product is reasonable is to estimate the product. Lucinda created greeting cards on her computer and sold them for $0.45 each. She sold 78 cards. How much money did she make?

Estimate 78 x 0.45

multiplication problem

Lucinda sold about $40.00 worth of cards.

Since both factors were rounded to a greater number, the actual product must be less than 40.

multiplication problem

Division by decimals is similar to division by whole numbers (covered earlier).

Rebecca enjoys pushing her scooter around the city-park pond after school and on weekends. If she circles the pond 7 times and travels a total distance of 5.88 km, what is the distance around the pond?

To divide a decimal by a whole number, divide the dividend, disregarding the decimal point. Then place the decimal point in the quotient directly above the decimal point in the dividend.

multiplication problem

When you divide a decimal by another decimal, there is one more step to the procedure.

Gloria spent Saturday participating in the local township Clean Scene day. She and her family walked along the rural roads picking up litter and trash that had accumulated over the winter. They cleaned a total of 4.2 miles of roadway. If they averaged about 0.6 miles of road per hour, how long did they spend picking up litter?

We want to know how many 0.6 mile are in 4.2 miles.
Find 4.2 ÷ 0.6.

Change the divisor to a whole number by mentally multiplying the divisor by a power of 10. Then multiply the dividend by the same power of 10 so the quotient will stay the same. As shown below, this mental math is accomplished by moving the decimal point one place to the right in the divisor and one place to the right in the dividend. Complete the division.

multiplication problem

Is 7 hours a reasonable the answer? 0.6 is about half a mile. 4.2 is about 4 miles. There are 8 half miles in 4 miles. The answer should be a little less than 8. Yes, 7 is reasonable.

Remind students that you can write zeros to the right of the last decimal place and not change the value of the decimal.

multiplication problem

Divide as though the dividend were a whole number. To continue the division, write 2 zeros (in the hundredths and thousandths places). Place the decimal point in the quotient directly above the decimal point in the dividend.

You may also need to place zeros in the dividend when dividing whole numbers to find the quotient to more decimal places.

multiplication problem

When 10 is multiplied by itself several times, you can make it easier to write by using an exponent. The exponent tells us how many times 10, the base, is used as a factor. 10 raised to an exponent is a power of 10.

10³ = 10 x 10 x 10 = 1,000
10² = 10 x 10 = 100
101 = 10

Notice the pattern between the exponent and the number of zeros in the standard form. The positive exponent of the power of ten will always be the same as the number of zeros in the standard form.

When you multiply a decimal by these powers of ten, notice what happens to the decimal point.

multiplication problems

2.435 x 101 = 24.35
2.435 x 10² = 243.5
2.435 x 10³ = 2,435

When you multiply a decimal by a power of 10, the decimal point moves to the right the number of places designated by the exponent. When you divide a decimal by a power of 10, the decimal point moves to the left—the opposite direction—the number of places designated by the exponent.

2.435 ÷ 101 = 0.2435
2.435 ÷ 10² = 0.02435

Many division problems require you to use the remainder in determining your answer.

The ecology club plans to serve ice-cream bars at their meeting. The club members need bars for 14 people. Bars come 4 in a package. What is the minimum number of packages they should purchase for their meeting?

multiplication problem

They need 3 full packages and another package for 2 more bars.If the club spends spent $9.76 for the ice cream, how much does each package cost?

multiplication problem

Each package costs $2.44.

Houghton Mifflin Math Grade 5