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Decimal Multiplication

Multiply decimals like whole numbers, then place the decimal point.

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Decimal Multiplication

Have you ever seen a dinosaur exhibit?

When the students in Mrs. Andersen’s class arrive at the Science Museum, Kara is very excited to learn that there is a dinosaur exhibit. In fact, it is a famous dinosaur exhibit. A set of dinosaur bones from a triceratops has been reconstructed and is on display. Kara can’t wait to get to see it. She has a feeling that this is going to be her favorite part of the museum. Several other students are equally excited, so Mrs. Andersen and the chaperones decide to go to the exhibit first and the split up into groups. When Kara walks in, she is delighted. There before her eyes is a huge skeleton of a triceratops. After visiting the exhibit for a while, the students begin to move on. Mrs. Andersen sees Kara hesitate before leaving the exhibit. She walks over to her.

“Imagine, that dinosaur is about 4 and a half times as long as you are!” Mrs. Andersen smiles. The students exit the exhibit hall, but Kara pauses at the door. She has to think about this. In all of her excitement she forgot to find the information that actually says how long the triceratops actually is. Mrs. Andersen’s words stay with her, “the dinosaur is 412\begin{align*}4 \frac{1}{2}\end{align*} times as long as you are.” Kara knows that she is 514\begin{align*}5 \frac{1}{4}\end{align*} feet tall. If the dinosaur is 412\begin{align*}4 \frac{1}{2}\end{align*} times as long as she is, how long is the dinosaur? While Mrs. Andersen and the chaperones start to split up the students, Kara begins working some quick math on the back of her museum map.

She writes down the following figures.

5.25 ×\begin{align*}\times\end{align*} 4.5 =\begin{align*}=\end{align*} ______

If Kara multiplies these numbers correctly, she will be able to figure out how long the triceratops is.

How long is he?

In this Concept you will learn all about multiplying decimals. When finished, you will know the length of the triceratops.

Problem 1:  Guidance

Sometimes, you will want to multiply two decimals without using a hundreds grid. You will want to use a method that is more efficient.

How can we multiply two decimals without using a hundreds grid?

Consider the problem 1.3 ×\begin{align*}\times\end{align*} .24.  We could solve the problem by

1. converting from decimals to mixed numbers or fractions (1.3×.24=1310×24100\begin{align*}1.3 \times .24 = 1\frac{3}{10} \times \frac{24}{100}\end{align*}),
2. performing fraction multiplication (1310×24100=1310×24100=3121000\begin{align*}1\frac{3}{10} \times \frac{24}{100} = \frac{13}{10} \times \frac{24}{100} = \frac{312}{1000}\end{align*}), and
3. converting back to a decimal (0.312),

but all that converting can become a nuisance.  Instead of doing all that work, we can use a shortcut.  Notice that the numerator in step 2 above is simply the two numbers multiplied together without any decimal points.

1.  Ignore the decimal points and multiply the two numbers as if the problem has two whole numbers (analogous to multiplying the two numerators).

The denominator is the power of ten from the first fraction times the power of ten from the second fraction.  The product of a power of ten and another power of ten is always a power of ten.  The number of zeros in the denominator of the first fraction is the same as the number of digits to the right of the decimal point in the first number.  The number of zeros in the denominator of the second fraction is the same as the number of digits to the right of the decimal point in the second number.  Therefore, the number of zeros in the product of the two denominators is the number of digits to the right of the decimal point in the product of our two numbers.   In shortcut form,

2.  Count the number of digits to the right of the decimal point in each factor (analogous to the number of zeros in each denominator).

3.  Add the number of digits to the right of the decimal points in each factor (analogous to multiplying the two denominators).

4.  Place the decimal point into the product from step 1 such that the product has the number of digits to the right of the decimal point as the sum found in step 3 (analogous to converting back from a fraction to a decimal number).

For 1.3 ×\begin{align*}\times\end{align*} .24,

1.

1.3×.2452+  260312\begin{align*}1.3 \\ \underline{\times \quad .24} \\ 52 \\ \underline{+ \ \ 260} \\ 312\end{align*}

2.  The first number has one decimal place:  1.3

The second number has two decimal places:  .24

3.  This is a total of three decimal places that need to be placed into the product.

4.  Our final answer is .312.

Problem 2:  How can we confirm our answer by using estimation?

When we multiply two decimals, sometimes we can use estimation to check our work.

4.7 ×\begin{align*}\times\end{align*} 2.1 =\begin{align*}=\end{align*} ______

We can start by rounding each decimal to the nearest whole number.

4.7 rounds to 5.

2.1 rounds to 2.

Next, we multiply 5 ×\begin{align*}\times\end{align*} 2 =\begin{align*}=\end{align*} 10.

Now let’s figure out our actual answer and see if our estimate is reasonable.

\begin{align*}4.7 \\ \underline{\times \quad \ 2.1} \\ 47 \\ \underline{+ \quad 940} \\ 9.87\end{align*}

We can see that our estimate is reasonable because 9.87 is very close to 10.

Practice

Now it is your turn. Multiply the following decimals.

Problem 3:

3.1 \begin{align*}\times\end{align*} 4.9 \begin{align*}=\end{align*} _____

Solution: 15.19

Problem 4:

1.2 \begin{align*}\times\end{align*} 5.1 \begin{align*}=\end{align*} _____

Solution: 6.12

Problem 5:

3.2 \begin{align*}\times\end{align*} 6.7 \begin{align*}=\end{align*} _____

Solution: 21.44

Now that you have learned all about multiplying decimals, let’s help Kara figure out the height of the triceratops. Here is the problem once again.

When the students in Mrs. Andersen’s class arrive at the Science Museum, Kara is very excited to learn that there is a dinosaur exhibit. In fact, it is a famous dinosaur exhibit. A set of dinosaur bones from a triceratops has been reconstructed and is on display. Kara can’t wait to get to see it. She has a feeling that this is going to be her favorite part of the museum. Several other students are equally excited, so Mrs. Andersen and the chaperones decide to go to the exhibit first and the split up into groups. When Kara walks in she is delighted. There before her eyes is a huge skeleton of a triceratops. After visiting the exhibit for a while, the students begin to move on. Mrs. Andersen sees Kara hesitate before leaving the exhibit. She walks over to her. “Imagine, that dinosaur is about 4 and a half times as long as you are!” Mrs. Andersen smiles.

The students exit the exhibit hall, but Kara pauses at the door. She has to think about this. In all of her excitement she forgot to find the information that actually says how tall the triceratops actually is.

Mrs. Andersen’s words stay with her, “the dinosaur is \begin{align*}4 \frac{1}{2}\end{align*} times as long as you are.”

Kara knows that she is \begin{align*}5 \frac{1}{4}\end{align*} feet tall. If the dinosaur is \begin{align*}4 \frac{1}{2}\end{align*} times as long as she is, how long is the dinosaur?

While Mrs. Andersen and the chaperones start to split up the students, Kara begins working some quick math on the back of her museum map.

She writes down the following figures.

5.25 \begin{align*}\times\end{align*} 4.5 \begin{align*}=\end{align*} ______

Let’s work on figuring out the height of the triceratops.

First, let’s estimate the product.

5.25 rounds down to 5.

4.5 rounds up to 5.

5 \begin{align*}\times\end{align*} 5 is 25 feet tall.

The triceratops is approximately 25 feet long.

Now let’s figure out its actual height.

\begin{align*}5.25 \\ \underline{\times \quad \ \ 4.5} \\ 2625 \\ \underline{+ \ \ 21000} \\ 23625 \end{align*}

Next, we add in the decimal point.

The triceratops is 23.6 feet long. He is a little longer than 23 and one-half feet.

Wow! That is one big dinosaur!!

Vocabulary

Hundreds grid
a grid of one hundred boxes used to show hundredths when working with decimals.
Product
the answer in a multiplication problem.
Vertically
written up and down in columns
Horizontally
written across

Review:

Try this problem on your own before checking the solution.

134 \begin{align*}\times\end{align*} .567 \begin{align*}=\end{align*} ______

Solution

This problem is going to have several steps to it because we are multiplying decimals that are in the thousandths place. That is alright though. We can do the same thing that we did with larger decimals. We can multiply the numbers as if they were whole numbers and then insert the decimal point at the end into the final product.

Let’s start by rewriting the problem vertically instead of horizontally.

\begin{align*}.134 \\ \underline{\times \quad \ .567} \\ 938 \\ 8040 \\ \underline{ + \ \ 67000} \\ 75978\end{align*}

Wow! There are a lot of digits in that number-now we need to put the decimal point into the product. There are three decimal places in the first number .134. There are three decimal places in the second number .567.

We need to count six decimal places from right to left in the product.

When this happens, we can add a zero in front of the digits to create the sixth place.

.075978

Further Practice

Directions: Multiply the following decimals.

1. 4.3 \begin{align*}\times\end{align*} .12 \begin{align*}=\end{align*} ______

2. 2.3 \begin{align*}\times\end{align*} 3.4 \begin{align*}=\end{align*} ______

3. .34 \begin{align*}\times\end{align*} .56 \begin{align*}=\end{align*} ______

4. 2.7 \begin{align*}\times\end{align*} 3.2 \begin{align*}=\end{align*} ______

5. 6.5 \begin{align*}\times\end{align*} 2.7 \begin{align*}=\end{align*} ______

6. .23 \begin{align*}\times\end{align*} .56 \begin{align*}=\end{align*} ______

7. 1.23 \begin{align*}\times\end{align*} .4 \begin{align*}=\end{align*} ______

8. .5 \begin{align*}\times\end{align*} .76 \begin{align*}=\end{align*} ______

9. .23 \begin{align*}\times\end{align*} .8 \begin{align*}=\end{align*} ______

10. 3.45 \begin{align*}\times\end{align*} 1.23 \begin{align*}=\end{align*} ______

11. 1.45 \begin{align*}\times\end{align*} .23 \begin{align*}=\end{align*} ______

12. .89 \begin{align*}\times\end{align*} .9 \begin{align*}=\end{align*} ______

13. .245 \begin{align*}\times\end{align*} .8 \begin{align*}=\end{align*} ______

14. 34.5 \begin{align*}\times\end{align*} .7 \begin{align*}=\end{align*} ______

15. 18.7 \begin{align*}\times\end{align*} .9 \begin{align*}=\end{align*} ______

16. 22.3 \begin{align*}\times\end{align*} .76 \begin{align*}=\end{align*} ______

17. 21.7 \begin{align*}\times\end{align*} .4 \begin{align*}=\end{align*} ______

18. 14.5 \begin{align*}\times\end{align*} .68 \begin{align*}=\end{align*} ______

19. 20.1 \begin{align*}\times\end{align*} .3 \begin{align*}=\end{align*} ______

20. 34.23 \begin{align*}\times\end{align*} .18 \begin{align*}=\end{align*} ______

21. .189 \begin{align*}\times\end{align*} .9 \begin{align*}=\end{align*} ______

22. .341 \begin{align*}\times\end{align*} .123 \begin{align*}=\end{align*} ______

23. .451 \begin{align*}\times\end{align*} .12 \begin{align*}=\end{align*} ______

24. .768 \begin{align*}\times\end{align*} .123 \begin{align*}=\end{align*} ______

25. .76 \begin{align*}\times\end{align*} .899 \begin{align*}=\end{align*} ______

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