Multiply a Decimal by a Whole Number | Grade 5 Math | LevelKite

You know that $6 \times 4$ means 6 equal groups of 4. A decimal works the same way: $6 \times 3.8$ means 6 equal groups of $3.8$ . The answer to a multiplication problem has a name: the product.

Before you find an exact product, it helps to know about how big it should be. An estimate is a number that is close to the exact answer and quick to find.

To estimate, round the decimal. Here that means you swap it for the nearest whole number. $3.8$ is between 3 and 4, and it is closer to 4. So 6 groups of $3.8$ is close to 6 groups of 4. Since $6 \times 4 = 24$ , the product $6 \times 3.8$ is about 24.

Rounding this way gives a useful estimate when the decimal is greater than 1, like every decimal in the table below. A decimal less than 1 can round to 0, and an estimate of 0 does not help you.

Three estimates. You will find each exact product later in this lesson.

Product to find	Round the decimal	Estimate
$6 \times 3.8$	$3.8$ is closer to $4$ than to $3$	$6 \times 4 = 24$ , so about $24$
$7 \times 1.94$	$1.94$ is closer to $2$ than to $1$	$7 \times 2 = 14$ , so about $14$
$13 \times 2.14$	$2.14$ is closer to $2$ than to $3$	$13 \times 2 = 26$ , so about $26$

An estimate tells you about how big a product is. A model is a picture of the numbers, and it lets you see the product exactly. Start with a small decimal, $0.4$ .

A tenth is one of 10 equal parts of a whole, and $0.4$ is 4 tenths. So $3 \times 0.4$ means 3 groups of 4 tenths.

The figure below has 2 bars. Each bar is 1 whole cut into 10 equal parts, so each part is 1 tenth. The figure shades 3 groups of 4 tenths, one after another. Count the shaded parts in fours: 4, 8, 12. That is $3 \times 4 = 12$ tenths. Ten of those tenths make 1 whole, and 2 tenths are left. So 12 tenths is 1 whole and 2 tenths, which you write as $1.2$ .

In the figure: 12 shaded tenths
In words: 3 groups of 4 tenths is 12 tenths
In numbers: $3 \times 0.4 = 1.2$

Each part is 1 tenth. Three groups of 4 tenths make 12 shaded tenths, so

3 \times 0.4 = 1.2

.

The same thing happens with smaller pieces. A hundredth is one of 100 equal parts of a whole, and $0.24$ is 24 hundredths. So $3 \times 0.24$ means 3 groups of 24 hundredths.

The grid below is 1 whole cut into 100 equal squares, so each small square is 1 hundredth. The grid shades 3 groups of 24 hundredths, one after another. That is $3 \times 24 = 72$ shaded hundredths. They do not fill the whole, so you write a 0 in the ones place: $0.72$ .

In the figure: 72 shaded hundredths
In words: 3 groups of 24 hundredths is 72 hundredths
In numbers: $3 \times 0.24 = 0.72$

Each small square is 1 hundredth. Three groups of 24 hundredths make 72 shaded hundredths, so

3 \times 0.24 = 0.72

.

Key concept

Multiply the count, keep the unit

In both models you only multiplied whole numbers: $3 \times 4 = 12$ and $3 \times 24 = 72$ . The size of the pieces never changed. Tenths stayed tenths, and hundredths stayed hundredths. The size of the piece you count has a name: the unit.

So to multiply a decimal by a whole number:

Name the decimal as a count of tenths or hundredths.
Multiply that count by the whole number.
Name the product in the same unit, then write it as a decimal.

The unit stays the same because a whole number only tells you how many groups there are. It does not cut the pieces any smaller.

Worked example

Tenths times a 1-digit number

Find $6 \times 3.8$ . Then check your product against the estimate from the table, about $24$ .

Name 3.8 as a count of tenths. Each whole is 10 tenths, so 3 wholes are 30 tenths. Add the 8 tenths, and 3.8 is 38 tenths.
$3.8 = 38 \text{ tenths}$
Multiply the count by 6, the way you multiply whole numbers, because 6 groups of 38 tenths is 6 × 38 tenths. Split 38 into 30 and 8. Then 6 × 30 = 180 and 6 × 8 = 48. Add the two parts.
$6 \times 38 = 180 + 48 = 228$
Name the product in the same unit: 228 tenths. Every 10 tenths make 1 whole, so 220 tenths make 22 wholes. The 8 tenths left over go in the tenths place.
$228 \text{ tenths} = 22.8$
Check the size with the estimate. 22.8 is close to 24, so the product is about the right size.

Answer

The product $6 \times 3.8$ is $22.8$ .

Worked example

Hundredths times a 1-digit number

Find $7 \times 1.94$ . Then check your product against the estimate from the table, about $14$ .

Name 1.94 as a count of hundredths. One whole is 100 hundredths, and 94 more hundredths make 194 hundredths.
$1.94 = 194 \text{ hundredths}$
Multiply the count by 7 as whole numbers, because 7 groups of 194 hundredths is 7 × 194 hundredths. Split 194 into 100, 90 and 4, multiply each part by 7, and add.
$7 \times 194 = 700 + 630 + 28 = 1358$
Name the product in the same unit: 1358 hundredths. Every 100 hundredths make 1 whole, so 1300 hundredths make 13 wholes, and 58 hundredths are left.
$1358 \text{ hundredths} = 13.58$
Check the size. 13.58 is close to the estimate of 14, so the product is about the right size.

Answer

The product $7 \times 1.94$ is $13.58$ .

A bigger whole number does not change the idea. $13 \times 2.14$ still means 13 equal groups of $2.14$ . You still name the decimal as a count, multiply the count, and keep the unit.

The whole-number multiplication now has more steps, so there are more places to slip. This is when an estimate helps most. It cannot prove that every digit is right. It does tell you about how big the product must be, so it catches a product that is far too big or far too small.

Worked example

A decimal times a 2-digit number

Find $13 \times 2.14$ . Use an estimate to check the size of your product.

Estimate first. 2.14 is closer to 2 than to 3, so round it to 2. Then 13 groups of 2 is 26. The product should be about 26.
$13 \times 2 = 26$
Name 2.14 as a count of hundredths. Two wholes are 200 hundredths, and 14 more hundredths make 214 hundredths.
$2.14 = 214 \text{ hundredths}$
Multiply the count by 13 as whole numbers. Split 13 into 10 and 3. Then 10 × 214 = 2140 and 3 × 214 = 642. Add the two parts.
$13 \times 214 = 2140 + 642 = 2782$
Name the product in the same unit: 2782 hundredths. Every 100 hundredths make 1 whole, so 2700 hundredths make 27 wholes, and 82 hundredths are left.
$2782 \text{ hundredths} = 27.82$
Check the size. 27.82 is close to the estimate of 26, so the product is about the right size. A product such as 278.2 would be far from 26, and you would know to look for a slip.

Answer

The product $13 \times 2.14$ is $27.82$ , which fits the estimate of about $26$ .

Here is the idea to keep. A decimal is a count of tenths or hundredths. When you multiply it by a whole number, you multiply the count and keep the unit. So $6 \times 3.8$ is 6 groups of 38 tenths, which is 228 tenths, or $22.8$ . When the decimal is greater than 1, rounding it to the nearest whole number gives you an estimate that tells you if your product is about the right size.

Next, you will multiply a decimal by another decimal. You will shade two bands on a hundredths grid, look at where they overlap, and find out what unit you get when tenths multiply tenths.