# 2.10: Decimal Multiplication

**At Grade**Created by: CK-12

**Practice**Decimal Multiplication

Peaches are Jackson’s favorite fruit. Each year he looks forward to peach season when he can go out and pick peaches at a local farm. He learns that this year, the peaches at the farm will cost $2.50 per pound. Jackson and his mom pick peaches until they fill up one large box. Their box filled with peaches weighs 15.8 pounds. How can Jackson calculate how much their box of peaches will cost?

In this concept, you will learn how to multiply decimals with and without rounding.

### Multiplying Decimals

Recall that in a decimal number, the **decimal point** divides the whole part of the number from the fractional part of the number.

Multiplying decimal numbers is very similar to multiplying whole numbers, but you will have to watch out for the decimal points. Remember that the answer to a multiplication problem is called the **product**.

To multiply decimal numbers:

- Arrange the numbers vertically so that the right-most digits for each number are lined up. Do not line up the decimal points, just the right-most digits.
- Temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.
- Count the total number of digits after the decimal points in the original numbers.
- In your answer, count the same number of places from right to left and insert a decimal point.

Here is an example.

Find the product of 3.25 and 1.2.

First, arrange the numbers vertically so that their right-most digits line up.

\begin{align*}\begin{array}{rcl}
&& \quad \ 3.25 \\
&& \underline{\;\;\; \times \ 1.2 \;\;}
\end{array}\end{align*}

Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.

\begin{align*}\begin{array}{rcl}
&& \quad \ \ 3.25 \\
&& \underline{\;\;\; \times \ 1.2 \;\;} \\
&& \quad \ \ \ 650 \\
&& \ \underline{ + \ 3250 \;\;} \\
&& \quad \ 3900
\end{array}\end{align*}

Now, count the total number of places after the decimal points in the original two numbers.

- 3.25 has 2 digits after the decimal point.
- 1.2 has 1 digit after the decimal point.

\begin{align*}2+1=3\end{align*}

Finally, count 3 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 3 and the 9.

\begin{align*}3.900\end{align*}

The answer is \begin{align*}3.25 \times 1.2=3.9\end{align*}

Sometimes you will want to round numbers before multiplying them. Rounding is useful when you only need an approximate answer instead of an exact answer.

Here is an example.

Round the numbers to the nearest hundredth and then find the product.

\begin{align*}3.748 \times 8.095\end{align*}

First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point.

- 3.748 rounds to 3.75
- 8.095 rounds to 8.10

Now, multiply the rounded numbers. Start by arranging the numbers vertically so that their right-most digits line up.

\begin{align*}\begin{array}{rcl}
&& \quad \ \ \ 3.75 \\
&& \underline{\;\;\times \ \ 8.10}
\end{array}\end{align*}

Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.

\begin{align*}
\begin{array}{rcl}
&& \qquad \ 3.75 \\
&& \underline{\;\;\; \times \ \ \ 8.10} \\
&& \qquad \quad \ \ 0 \\
&& \quad \quad 3750 \\
&& \ \underline{+300000} \\
&& \quad 303750
\end{array}\end{align*}

Now, count the total number of places after the decimal points in the original two numbers. There are 4 digits total after the decimal points in the original numbers.

Finally, count 4 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 0 and the 3.

\begin{align*}30.3750\end{align*}

The answer is \begin{align*}3.75 \times 8.10=30.375\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Jackson and his peaches.

Jackson and his mom picked 15.8 pounds of peaches and the peaches cost $2.50 per pound. Jackson needs to calculate how much their peaches will cost.

To figure out the total cost, Jackson needs to multiply 2.50 by 15.8.

First, Jackson should arrange the numbers vertically so that their right-most digits line up.

\begin{align*}\begin{array}{rcl}
&& \quad \ \ \ 2.50\\
&& \underline{\;\; \times \ \ 15.8} \\
\end{array}\end{align*}

Next, he should temporarily ignore the decimal points and multiply just as if he were multiplying whole numbers.

\begin{align*}
\begin{array}{rcl}
&& \qquad 2.50 \\
&& \underline{\;\; \times \ \ \ 15.8} \\
&& \quad \ \ \ \ 2000 \\
&& \quad \ \ 12500 \\
&& \ \ \underline{\; +25000} \\
&& \quad \ \ 39500
\end{array}
\end{align*}

Now, he should count the total number of places after the decimal points in the original two numbers. There are 3 digits total after the decimal points in the original numbers.

Finally, he should count 3 places from right to left in the answer and insert a decimal point. He will place the decimal point between the 9 and the 5.

\begin{align*}39.500\end{align*}

The answer is the peaches will cost $39.50.

#### Example 2

Round the numbers to the nearest tenth and then multiply.

\begin{align*}5.68 \times 1.38\end{align*}

First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.

- 5.68 rounds to 5.7
- 1.38 rounds to 1.4

Now, multiply the rounded numbers. Start by arranging the numbers vertically so that their right-most digits line up.

\begin{align*}\begin{array}{rcl}
&& \ \ \ \ \ 5.7 \\
&& \underline{\; \times \ 1.4}
\end{array}\end{align*}

Next, temporarily ignore the decimal points and multiply just as if you were multiplying whole numbers.

\begin{align*}\begin{array}{rcl}
&& \quad \ 5.7 \\
&& \underline{\; \times \ 1.4} \\
&& \quad 228 \\
&& \ \underline{ +570} \\
&& \quad 798
\end{array}\end{align*}

Now, count the total number of places after the decimal points in the original two numbers. There are 2 digits total after the decimal points in the original numbers.

Finally, count 2 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 7 and the 9.

\begin{align*}7.98\end{align*}

The answer is \begin{align*}5.7 \times 1.4=7.98\end{align*}

#### Example 3

Find the product of 1.23 and 6.7.

First, arrange the numbers vertically so that their right-most digits line up.

\begin{align*}\begin{array}{rcl}
&& \quad \ 1.23 \\
&& \underline{ \times \ \ \ \ 6.7} \\
\end{array}\end{align*}

\begin{align*}\begin{array}{rcl}
&& \quad \ 1.23 \\
&& \underline{ \times \ \ \ \ 6.7} \\
&& \quad \ \ 861 \\
&& \ \underline{+7380} \\
&& \quad 8241
\end{array}\end{align*}

Now, count the total number of places after the decimal points in the original two numbers.

- 1.23 has 2 digits after the decimal point.
- 6.7 has 1 digit after the decimal point.

\begin{align*}2+1=3\end{align*}

Finally, count 3 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 8 and the 2.

\begin{align*}8.241\end{align*}

The answer is \begin{align*}1.23 \times 6.7=8.241\end{align*}

#### Example 4

Find the product of 4.56 and 1.34.

First, arrange the numbers vertically so that their right-most digits line up.

\begin{align*}
\begin{array}{rcl}
&& \quad 4.56 \\
&& \underline{\times \ 1.34} \\
\end{array}\end{align*}

\begin{align*}\begin{array}{rcl}
&& \quad \ \ \ 4.56 \\
&& \underline{\; \times \ \ \ 1.34} \\
&& \quad \ \ 1824 \\
&& \quad 13680 \\
&& \ \underline{+45600} \\
&& \quad 61104
\end{array}\end{align*}

Now, count the total number of places after the decimal points in the original two numbers. There are 4 digits total after the decimal points in the original numbers.

Finally, count 4 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 6 and the 1.

\begin{align*}6.1104\end{align*}

The answer is \begin{align*}4.56 \times 1.34=6.1104\end{align*}

#### Example 5

Round the numbers to the nearest tenth and then multiply.

\begin{align*}5.67 \times 4.35\end{align*}

First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point.

- 5.67 rounds to 5.7
- 4.35 rounds to 4.4

Now, multiply the rounded numbers. Start by arranging the numbers vertically so that their right-most digits line up.

\begin{align*}
\begin{array}{rcl}
&& \quad \ 5.7\\
&& \underline{ \times \ \ 4.4} \\
\end{array}
\end{align*}

\begin{align*}\begin{array}{rcl}
&& \quad \ \ 5.7\\
&& \underline{ \times \ \ \ 4.4} \\
&& \quad \ \ 228 \\
&& \ \underline{+2280} \\
&& \quad 2508
\end{array}\end{align*}

Now, count the total number of places after the decimal points in the original two numbers. There are 2 digits total after the decimal points in the original numbers.

Finally, count 2 places from right to left in the answer and insert a decimal point. You will place the decimal point between the 5 and the 0.

\begin{align*}25.08\end{align*}

The answer is \begin{align*}5.7 \times 4.4=25.08\end{align*}

### Review

Find the products.

- \begin{align*}12.7\times0.8\end{align*}
12.7×0.8 - \begin{align*}0.552\times0.3\end{align*}
0.552×0.3 - \begin{align*}6.09\times3.34\end{align*}
6.09×3.34 - \begin{align*}25.6\times0.72\end{align*}
25.6×0.72 - \begin{align*}56.71\times.34\end{align*}
56.71×.34 - \begin{align*}.45\times4.3\end{align*}
.45×4.3 - \begin{align*}1.234\times7.8\end{align*}
1.234×7.8

Find the product after rounding each decimal to the nearest tenth.

- \begin{align*}33.076\times5.228\end{align*}
33.076×5.228 - \begin{align*}9.29\times0.6521\end{align*}
9.29×0.6521 - \begin{align*}4.5513\times4.874\end{align*}
4.5513×4.874 - \begin{align*}12.48\times7.95\end{align*}
12.48×7.95 - \begin{align*}14.56\times4.52\end{align*}
14.56×4.52 - \begin{align*}8.76\times1.24\end{align*}
8.76×1.24 - \begin{align*}9.123\times6.789\end{align*}
9.123×6.789 - \begin{align*}9.3323\times8.719\end{align*}
9.3323×8.719

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.10.

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In this concept, you will learn how to multiply decimals with and without rounding.

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