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Estimating
Decimal Sums |
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Unit 12 > Lesson 5 of 12 |
| Example 1: |
Charlene and Margi went to a nearby restaurant for a quick lunch. The
waitress handed them the bill with the price of each lunch, but forgot to add up the total. Estimate the total bill if Charlene's lunch
was $3.75 and Margi's lunch was $4.29.
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| Analysis: |
Estimation is a good tool for making a rough calculation. There are many estimation strategies that you could use to estimate the sum of
these decimals. Let's look at the front-end strategy and the rounding strategy.
We will round to the nearest tenth. |
| Estimates: |
| Front-End Strategy |
|
Rounding Strategy |
| Add the front digits and then adjust the estimate. |
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Round each decimal to a designated place value, then add to
estimate the sum. |
3.75
3.75
about
+ 4.29
+ 4.29
1
7
7 + 1 = 8 |
|
3.75
3.8
+ 4.29+
4.3
+ 4.29 +
8.1 |
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| 3 + 4 = 7 and .75 plus .29 is about 1. Thus, $7+ $1 = $8. The
estimated sum is $8. |
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Rounding each decimal to the nearest tenth, we get an
estimated sum of 8.1 or $8.10. |
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| Answer: |
Both $8 and $8.10 are good estimates for the total lunch
bill for Charlene and Margi.
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Note the difference between the two strategies used in Example 1: The
front-end strategy uses the first digit to estimate a sum and then
considers the other digits to adjust the estimate. In the rounding strategy,
addition does not occur until after the numbers have been rounded. In both
strategies, you must line up both decimals before proceeding.
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| Procedure: |
To round a decimal to a designated place value, first underline or mark that place. If the digit to the right of that place is 5 through 9, then round up. If the digit to the right of that place is 1 through 4, then round down,
leave the digit in the designated place unchanged, and drop all digits to the right of it. |
Let's look at some more examples of estimating decimal sums.
| Example 2: |
Estimate the sum of each pair of decimals by rounding to the specified place.
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| a) |
Estimate 4.203 + 6.598 by rounding to the nearest hundredth.
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4.203
4.20
+ 6.598+
6.60
10.80
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| If the digit to the right of the place you are
rounding to is 1 through 4, then round down. Thus, 4.203 is rounded
down to 4.20. |
| If the digit to the right of the place you are
rounding to is 5 through 9, then round up. Thus, 6.598 is rounded up to
6.60. |
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| b) |
Estimate $12.96 + $7.19 by rounding to the nearest one.
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$12.96
$13
+ $ 7.19+
$ 7
$20
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| If the digit to the right of the place you are
rounding to is 5 through 9, then round up. Thus, 12.96 is rounded up to
13. |
| If the digit to the right of the place you are
rounding to is 1 through 4, then round down. Thus, 7.19 is rounded
down to
7. |
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| c) |
Estimate 11.79 + 4.58 by rounding to the nearest tenth.
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11.79 11.8
+ 4.58+
4 6
16.4
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| If the digit to the right of the place you are
rounding to is 5 through 9, then round up. Thus, 11.79 is rounded up to
11.8. |
| If the digit to the right of the place you are
rounding to is 5 through 9, then round up. Thus, 4.58 is rounded up to
4.6. |
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| Example 4: |
Use either method to estimate the sum of 6.37 + 2.8 + 21.19
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Answer:
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| Front-End Strategy |
|
Rounding Strategy |
6.37
6.37
about
2.80
2.80
.
+ 21.19
+ 21.19
1
29
29
+ 1 = 30 |
|
6.37
6.4
2.80
2.8
+ 21.19+
21.2
30.4 |
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| 6 + 2 + 21 = 29 and.37 + .80 + .19 is about 1. Thus, 29 + 1 = 30. The
estimated sum is 30. |
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Rounding each decimal to the nearest tenth, we get an
estimated sum of 30.4. |
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As you can see from
our examples, estimates will vary depending on which strategy is used. In Example 4, we got an estimated sum of 30 using the front-end strategy, and an
estimate of 30.4 by rounding to the nearest tenth. The actual sum is 30.36, thus
30 is an underestimate and 30.4 is an overestimate.
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| Definition 1: |
An overestimate is an an estimate that is too high:
it exceeds the actual answer. |
| Definition 2: |
An underestimate is an estimate that is too low:
it is lower than the actual answer. |
| Example 5: |
Estimate the sum of each pair of decimals by rounding to the nearest tenth.
Then indicate if the estimate is an overestimate or an underestimate by
comparing it to the actual answer.
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Estimate
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Actual Sum
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Overestimate or Underestimate?
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| a) |
+ $36.23+
$36.20
+ $63.44+
$63.40
+ $99.60+
$99.60
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$99.67
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Underestimate since $99.60 < $99.67 |
| b) |
$57.65
$ 57.70
+ $43.31+ $
43.30
$101.00
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$100.96
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Overestimate since $101.00 > $100.96 |
| c) |
$24.89
$24.90
+ $72.15+ $72.20
$97.10
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$97.04
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Overestimate since $97.10 > $97.04 |
| Example 6: |
A marble rolled 3.9658 cm and then rolled 16.37 cm. Jen estimated that the
marble rolled 20 cm altogether. If the actual sum is 20.3358 cm, did she
overestimate or underestimate? Explain your answer.
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| Answer: |
Jen underestimated because her estimate of 20 cm was
lower than the actual sum of 20.3358 cm.
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| Summary: |
Estimation is a good tool for making a rough calculation. In this lesson, we learned how to estimate decimal sums using two different
strategies: front-end and rounding. In both
strategies, you must line up all decimals before proceeding. An overestimate is an an estimate that is too
high; an underestimate is an estimate that is too low.
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Exercises
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In Exercises 1 through 3, you may use paper and pencil to help you estimate.
Click once
in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to
indicate whether your answer is correct or incorrect. To start over, click CLEAR.
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1.
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Estimate the sum of 0.79 and 0.13 by rounding to the nearest tenth.
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2.
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Estimate the sum of 3.197 and 4.214 by rounding to the nearest hundredth.
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3.
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Estimate the sum of 1.768 and 2.209 using the front-end strategy.
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In Exercises 4 and 5, read each question below. Select your answer by clicking
on its button. Feedback to your answer is provided in the RESULTS BOX. If you
make a mistake, choose a different button.
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4.
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The sum of two decimals is 56.9531. If Aaron overestimated the
sum, then which of the following estimates did he use?
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5.
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The sum of two decimals is 31.5487. If Lana underestimated the
sum, then which of the following estimates did she use?
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