Zachary and Gregory are marine biologists. Their research involves sea trout. They are using a submersible that dives at a rate of 8 feet every 30 seconds. How deep will the submersible dive in 3 minutes, 5 minutes, and 10 minutes? How long will it take the submersible to reach a depth of 3,280 feet?

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

### Multiplying and Dividing Decimals

You can multiply decimals in the same way that you multiply whole numbers with several digits. When you multiply decimals, just follow these two steps:

First, multiply normally, ignoring the decimal points.

Next, put the decimal point in the answer. The answer will have as many decimal places as the original numbers combined.

Let’s look at an example.

Multiply:

First, line up the numbers in order to get ready to multiply.

Next, multiply each digit in the top number by each digit in the bottom number, just like whole numbers.

Then, place the decimal point in the product by counting the number of decimal places in each of the numbers that were multiplied. The first number has two decimal places, and the second number has one decimal place. So move the decimal point three places.

The answer is 284.294.

When you divide decimals, you have to pay attention to the decimal point. There are also two steps for dividing decimals.

First, use division or long division ignoring the decimal point.

Next, put the decimal point in the same spot as the **dividend** (the number being divided).

Note that the **dividend** is the number being divided so it goes into our division box and the **divisor** is the number doing the dividing.

Let’s look at an example.

Divide:

First, set up the long division.

Next, ignore the decimal point and divide as you would divide whole numbers.

The dividend is 2532.6.

Then, place the decimal point in the quotient directly above the decimal point in the dividend.

The answer is 56.28.

To estimate products and quotients with decimals, you need to first round the numbers so that they are easier to work with. To round to the nearest whole number, look at the digit in the tenths place. If it is less than 5, round down. If it is 5 or greater, round up.

Remember that an estimate is an answer that is not exact, but is approximate and reasonable.

Let’s look at an example.

Estimate the product:

First, round the first number. Since there is a 2 in the tenths place, 11.256 rounds down to 11.

Next, round the second number. Since there is an 8 in the tenths place, 6.81 rounds up to 7.

Then, multiply the rounded numbers.

The answer is 77.

This is an estimate of the product:

.Let’s look at an example where you need to estimate using division.

Estimate the quotient:

First, round the first number. Since there is a 9 in the tenths place, 91.93 rounds up to 92.

Next, round the second number. Since there is a 3 in the tenths place, 4.39 rounds down to 4.

Then, divide the rounded numbers.

The answer is 23.

This is an estimate of the quotient:

.### Examples

#### Example 1

Earlier, you were given a problem about Zachary's and Gregory's submersible dive.

They are trying to determine how deep their submersible will dive in 3 minutes, in 5 minutes, and in 10 minutes. As well, they want to find out how long it will take the submersible to dive to 3280 feet. Can you help them?

You know the submersible dives 8 feet every 30 seconds which means it dives at 16 feet every minute.

For 3 minutes, the depth would be \begin{align*}16 \times 3\end{align*} or 48 feet.

The answer is 48 feet.

For 5 minutes, the depth would

be or 80 feet.The answer is 80 feet.

For 10 minutes, the depth would

be or 160 feet.The answer is 160 feet.

Now, how long will it take the submersible to reach 3280 feet?

You need to divide the depth by the rate the submersible dives.

First, set up the long division.

\begin{align*}{16 \overline{ ) {3280}}}\end{align*}

Next, since there are no decimal points, simply perform the long division.

The answer is 205 minutes.

#### Example 2

Solve the following problem.

Angus bought five new telephones for the school office. They cost $61.35 each. If the price includes tax, estimate how much Angus spent? What is the exact amount that Angus spent?

Let's start with the estimation.

First, round the price of each phone to a number that is easy to multiply. 61.35 would round to 61.

Next, multiply by 5.

The answer is 305 so Angus spent approximately $305.

This is the estimate.

Now let's find the exact amount.

First, line up the numbers in order to get ready to multiply.

Next, multiply each digit in the top number by each digit in the bottom number, just like whole numbers.

Then, place the decimal point in the product by counting the number of decimal places in each of the numbers that were multiplied. The first number has two decimal places, and the second number has no decimal places. So move the decimal point two places.

The answer is $306.75.

#### Example 3

First, set up the long division.

Next, ignore the decimal point and divide as you would divide whole numbers.

\begin{align*}\begin{array}{rcl} && \overset{ \quad \ \ 35}{2.15 \overline{ ) {75.25}}}\\ && \qquad \ \underline{645}\\ && \qquad \ 1075\\ && \qquad \ \underline{1075}\\ && \qquad \quad0 \end{array}\end{align*}

Then, place the decimal point in the quotient directly above the decimal point in the dividend.

The answer is 35.

#### Example 4

First, set up the long division.

Next, ignore the decimal point and divide as you would divide whole numbers.

Then, place the decimal point in the quotient directly above the decimal point in the dividend.

The answer is 6.9.

#### Example 5

First, line up the numbers in order to get ready to multiply.

\begin{align*}14.50\\ \underline{\times \;\;\; 2.1}\end{align*}

Next, multiply each digit in the top number by each digit in the bottom number, just like whole numbers.

\begin{align*}& 14.50\\ & \underline{\times \; 2.1}\\ & \ \ 1450\\ & \underline{29000}\\ & 30450\end{align*}

Then, place the decimal point in the product by counting the number of decimal places in each of the numbers that were multiplied. The first number has two decimal places, and the second number has one decimal place. So move the decimal point three places.

The answer is 30.45.

### Review

Estimate each product using rounding.

- \begin{align*}2.67 \times 3.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}4.15 \times 8.09 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6.67 \times 7.10 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}8.21 \times 9.87 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5.86 \times 5.13 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}7.24 \times 4.63 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6.35 \times 12.01 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}4.13 \times 9.87 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}8.12 \times 9.15 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}16.21 \times 9.94 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Estimate each quotient using rounding.

- \begin{align*}21.87 \div 2.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}32.14 \div 8.03 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}36.07 \div 8.83 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}16.20 \div 7.92 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}34.87 \div 5.03 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}18.08 \div 3.14 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}21.10 \div 3.17 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}44.82 \div 8.60 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}120.02 \div 58.72 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}139.87 \div 69.81 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Multiply or divide to find each product or quotient.

- \begin{align*}13.64 \div 2.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}21.35 \div 6.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5.2 \times 6.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6.7 \times 4.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}0.437 \times 2.1 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.3.