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2.11: Estimation to Check Decimal Multiplication

Difficulty Level: At Grade Created by: CK-12

Have you ever wondered about high school track records? Travis has been doing some research.

In his preparation for running the 440, Travis began to research national track records. On July 24, 1982, Darrell Robinson from Wilson High School in Tacoma Washington ran the 440 in 44.69. Travis is amazed by the time and ready to work on his own personal best.

If Darrell ran this 1.5 times, what would the total time be?

You can solve this problem by multiplying leading digits. Pay attention and you will understand how to do this by the end of the Concept.

Guidance

With addition and subtraction of decimals, you have seen how estimation works to approximate a solution. It is a good idea to get in the habit of estimating either before or after solving a problem. Estimation helps to confirm that your solution is in the right ballpark.

With multiplication, rounding decimals before multiplying is one way to find an estimate. You can also simply multiply the leading digits.

Remember how we used front-end estimation to approximate decimal sums and differences? Multiplying leading digits works the same way. The leading digits are the first two values in a decimal. To estimate a product, multiply the leading digits exactly as you have been—align decimals to the right, then insert the decimal point into the solution based on the sum of places in the original numbers.

Estimate the product, \begin{align*}6.42 \times 0.383\end{align*}.

First we need to identify the leading digits, being careful to preserve the placement of the decimal point in each number. In 6.42, the leading digits are 6.4. In 0.383, the leading digits are .38. Note that 0 is not one of the leading digits in the second decimal. Because zero is the only number on the left side of the decimal point, we can disregard it. Now that we have the leading digits, we align the decimals to the right and multiply.

\begin{align*}& \quad \ \ 6.4\\ & \ \ \underline{\times \; .38\ }\\ & \quad \ \ 512\\ & \underline{+ \; 1920 \ }\\ & \quad 2.432\end{align*}

Find the product. Then estimate to confirm your solution. \begin{align*}22.17 \times 4.45\end{align*}.

This problem asks us to perform two operations—straight decimal multiplication followed by estimation. Let’s start multiplying just as we’ve learned: aligning the decimals to the right and multiplying as if they were whole numbers.

\begin{align*}& \quad \ \ \ 22.17\\ & \quad \ \underline{\times \; 4.45 \ }\\ & \quad \ \ 11085\\ & \quad \ \ 88680\\ & \underline{+ \; 886800 \ }\\ & \ \ 98.6565\end{align*}

Note the placement of the decimal point in the answer. The original factors both have two decimal places, so once we have our answer, we count over four decimal places from the right, and place the decimal point between the 8 and the 6.

Now that we have multiplied to find the answer, we can use estimation to check our product to be sure that it is accurate. We can use the method multiplying leading digits. First, we reduce the numbers to their leading digits, remaining vigilant as to the placement of the decimal point. The leading digits of 22.17 are 22. The leading digits of 4.45 are 4.4. Now we can multiply.

\begin{align*}& \quad \ 22\\ & \underline{\times \; 4.4 \ }\\ & \quad \ 88\\ & \underline{+ \; 880 \ }\\ & \ \ 96.8\end{align*}

If you look at the two solutions, 98.6 and 96.8, you can see that they are actually very close. Our estimate is very close to the actual answer. We can trust that our answer is accurate.

Find each product. Use multiplying leading digits to check your answer.

Example A

\begin{align*}67.9 \times 1.2\end{align*}

Solution:\begin{align*}80.4\end{align*}

Example B

\begin{align*}5.321 \times 2.301\end{align*}

Solution: \begin{align*}12.19\end{align*}

Example C

\begin{align*}8.713 \times 9.1204\end{align*}

Solution: \begin{align*}79.17\end{align*}

Here is the original problem once again.

In his preparation for running the 440, Travis began to research national track records. On July 24, 1982, Darrell Robinson from Wilson High School in Tacoma Washington ran the 440 in 44.69. Travis is amazed by the time and ready to work on his own personal best.

If Darrell ran this 1.5 times, what would the total time be?

To figure this out, we can first write a problem showing only the leading digits.

\begin{align*}44 \times 1.5\end{align*}

Next, we multiply.

\begin{align*}66\end{align*}

If Darrell had run this distance 1.5 times, his total time would be about 66 minutes.

Vocabulary

Here are the vocabulary words in this Concept.

Product
the answer in a multiplication problem.
Estimation
finding an approximate answer through rounding or multiplying leading digits

Guided Practice

Here is one for you to try on your own.

\begin{align*}.4561 \times .32109\end{align*}

Answer

To multiply by using leading digits, we need to identify the leading digits first.

\begin{align*}.45\end{align*}

\begin{align*}.32\end{align*}

Now we multiply.

\begin{align*}.45 \times .32 = .144\end{align*}

This is our answer.

Video Review

Here is a video for review.

- This is a Khan Academy video on multiplying decimals.

Practice

Directions: Estimate the products by multiplying the leading digits.

1. \begin{align*}7.502 \times 0.9281\end{align*}

2. \begin{align*}46.14 \times 2.726\end{align*}

3. \begin{align*}0.39828 \times 0.16701\end{align*}

4. \begin{align*}83.243 \times 6.517\end{align*}

5. \begin{align*}5.67 \times .987\end{align*}

6. \begin{align*}7.342 \times 1.325\end{align*}

7. \begin{align*}17.342 \times .325\end{align*}

8. \begin{align*}.34291 \times 1.525\end{align*}

9. \begin{align*}.5342 \times .87325\end{align*}

10. \begin{align*}.38942 \times .9825\end{align*}

11. \begin{align*}7.567 \times 3.325\end{align*}

12. \begin{align*}12.342 \times 11.325\end{align*}

13. \begin{align*}21.342 \times 14.555\end{align*}

14. \begin{align*}.110342 \times .098325\end{align*}

15. \begin{align*}37.1342 \times 1.97325\end{align*}

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Date Created:
Dec 21, 2012
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Feb 26, 2016
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