Multiply Two Decimals | Grade 5 Math | LevelKite

You already know how to multiply a decimal by a whole number. For example, $3 \times 0.7$ means 3 groups of $0.7$ . In this lesson both factors (the numbers you multiply) are decimals.

What does $0.3 \times 0.7$ mean? It means $0.3$ of $0.7$ . You do not take a whole group of $0.7$ . You take only 3 tenths of it. The answer to a multiplication is called the product, so this product is a part of a part.

A hundredths grid lets you see a part of a part. It is one whole square cut into 10 equal rows and 10 equal columns. That makes $10 \times 10 = 100$ equal small squares, so each small square is one hundredth of the whole. A full row is 10 small squares, which is one tenth of the whole. So is a full column.

On the grid that comes next, two bands are shaded. A band is a strip of full rows or full columns. The band of 7 columns shows $0.7$ , because 7 of the 10 columns are 7 tenths. The band of 3 rows shows $0.3$ , because 3 of the 10 rows are 3 tenths. The overlap is the part that lies inside both bands.

A band of 3 rows shows

0.3

and a band of 7 columns shows

0.7

. The overlap is 3 rows by 7 columns.

Worked example

Find a product on a hundredths grid

Use the hundredths grid just before this example to find $0.3 \times 0.7$ .

Find each factor on the grid. The band of 7 columns covers 7 of the 10 equal columns, so it shows 7 tenths, or 0.7. The band of 3 rows covers 3 of the 10 equal rows, so it shows 3 tenths, or 0.3.
$0.7 = 7 \text{ tenths}, \quad 0.3 = 3 \text{ tenths}$
Look at the overlap, the small squares inside both bands. The 10 rows cut the band of 7 columns into 10 equal parts. The overlap is 3 of those parts, so it is 3 tenths of that band. That makes the overlap 0.3 of 0.7, which is the product you want.
Count the small squares in the overlap. The overlap is 3 rows by 7 columns, so you can multiply the whole numbers instead of counting one square at a time.
$3 \times 7 = 21$
Name what you counted. Each small square is one hundredth of the whole, so 21 small squares are 21 hundredths. Write 21 hundredths as the decimal 0.21.
$0.3 \times 0.7 = 0.21$

Answer

The product of $0.3$ and $0.7$ is $0.21$ . On the grid, it is the 21 small squares where the two bands overlap, which is 21 hundredths.

The grid also shows why the product is in hundredths. One row is one tenth of the whole. One column is one tenth of the whole. One row and one column cross in exactly one small square, and that square is one hundredth. So one tenth of one tenth is one hundredth.

Look at what you did in the example. You multiplied the whole numbers, $3 \times 7 = 21$ , to count the pieces. Then you named the size of each piece, which is called the unit. Here the unit is hundredths. In words, 3 tenths × 7 tenths are 21 hundredths. Tenths × tenths are hundredths.

Sometimes one factor is in hundredths, as in $0.6 \times 0.24$ . What unit do you get then? Picture one small square of the grid, which is one hundredth. Cut it into 10 equal slices and keep one slice. That slice is one tenth of one hundredth.

How big is the slice? It takes 10 slices to fill a small square and 100 small squares to fill the whole. So $10 \times 100 = 1000$ slices fill the whole, and each slice is one thousandth of the whole. Tenths × hundredths are thousandths.

Key concept

Multiply the whole numbers, then name the unit

To multiply two decimals, say each factor in its unit, such as 6 tenths or 24 hundredths. Multiply the whole numbers to find how many pieces there are. Then name how big each piece is:

tenths × tenths are hundredths
tenths × hundredths are thousandths

The unit tells you where the decimal point goes in the product.

0.1 \times 0.1 = 0.01 \qquad 0.1 \times 0.01 = 0.001

Worked example

Tenths times hundredths

Find $0.6 \times 0.24$ .

Say each factor in its unit. The last digit of 0.6 is in the tenths place, so 0.6 is 6 tenths. The last digit of 0.24 is in the hundredths place, so 0.24 is 24 hundredths.
$0.6 \times 0.24 = 6 \text{ tenths} \times 24 \text{ hundredths}$
Multiply the whole numbers 6 and 24 to find how many pieces there are. Split 24 into 20 and 4, multiply each part by 6, and add.
$6 \times 24 = 6 \times 20 + 6 \times 4 = 120 + 24 = 144$
Name the unit. Tenths × hundredths are thousandths, so the product is 144 thousandths.
$6 \text{ tenths} \times 24 \text{ hundredths} = 144 \text{ thousandths}$
Write 144 thousandths as a decimal. The last digit must sit in the thousandths place, which is the third place after the decimal point.
$0.6 \times 0.24 = 0.144$

Answer

The product of $0.6$ and $0.24$ is $0.144$ , which is 144 thousandths.

A band of 3 rows and a band of 2 columns overlap in 6 small squares. That is 6 hundredths, so

0.3 \times 0.2 = 0.06

.

You can also place the decimal point by thinking about size. The whole-number multiplication gives you the digits of the product. The decimal point decides how big the number is.

So make an estimate, a quick answer that is close but not exact. When the factors are greater than 1, round each one to a whole number and multiply. The real product will be about that size. Moving the decimal point one place makes a number ten times as big or one tenth as big, so only one place fits your estimate.

Then check with the units. When the estimate and the unit name put the decimal point in the same place, you can trust your answer.

Worked example

Place the decimal point two ways

Find $2.4 \times 3.6$ . Place the decimal point by thinking about size, then check with units.

Multiply as if the factors were the whole numbers 24 and 36. This gives the digits of the product. Split 36 into 30 and 6, multiply each part by 24, and add.
$24 \times 36 = 24 \times 30 + 24 \times 6 = 720 + 144 = 864$
Estimate the size. The factor 2.4 is close to 2, and 3.6 is close to 4, so the product is about 2 × 4.
$2 \times 4 = 8$
Place the decimal point so the digits 864 make a number near 8. The choices include 0.864, 8.64 and 86.4. Only 8.64 is near 8. The number 0.864 is about one tenth of 8, and 86.4 is about ten times 8.
$2.4 \times 3.6 = 8.64$
Check with units. The factor 2.4 is 24 tenths and 3.6 is 36 tenths. Tenths × tenths are hundredths, so the product is 864 hundredths. Since 100 hundredths make 1 whole, that is 8 wholes and 64 hundredths, or 8.64. Both ways agree.
$24 \text{ tenths} \times 36 \text{ tenths} = 864 \text{ hundredths} = 8.64$

Answer

The product of $2.4$ and $3.6$ is $8.64$ . The estimate of about 8 and the unit name, 864 hundredths, put the decimal point in the same place.

Takeaway. To multiply two decimals, multiply the whole numbers to count the pieces. Then name the size of the pieces: tenths × tenths are hundredths, and tenths × hundredths are thousandths. On a hundredths grid, the product is the overlap of two bands. You can also place the decimal point by estimating the size of the product. The two ways should agree.

Next, you will solve word problems that multiply decimals, compare a product with one of its factors, and find and fix mistakes.