# Estimating Decimal Differences

Example 1: A customer wants to buy \$6.33 of food in a store. About how much change will he receive from a \$10 bill if the cashier has no pennies and no nickels?

Analysis: Estimate the difference between \$10.00 and \$6.33 by rounding to the nearest tenth (dime).

\$10.00\$10.00
- \$  6.33- \$  6.30
- \$  6.33\$  3.70

Answer: The customer will receive about \$3.70 in change from the cashier.

Estimation is a good tool for making a rough calculation. It is also used to determine if an answer is reasonable. For example, if the cashier had given the customer 40 cents (\$0.40), he could have used estimation to identify her mistake: The customer would make a rough calculation and realize that the cashier is off by a factor of 10. This is an important life skill to have! Without the ability to estimate, a consumer can get short-changed! Let's look at some more examples of estimating decimal differences.

Example 2: Two students estimated the difference of these decimals: 18.32 - 4.689 as shown below. Which student's estimate was reasonable? Explain your answer.

Student 1:

- 18.320- 20

- 04.689-   0

- 18.32020

Student 1 rounded to the nearest ten and got an estimated difference of 20.

Student 2:

- 18.320- 18

-   4.689-   5

- 18.32013

Student 2 rounded to the nearest one and got an estimated difference of 13.

Answer: Student 1 had an estimate of 20, and 20 is greater than 18.32. This estimate is unreasonable since it is more than either of the original numbers. When estimating a difference, the estimate should not exceed the original numbers. Student 2 had a reasonable estimate since it did not exceed the original numbers.

Note that in the examples above, estimation is used to determine if an answer is reasonable, not to find an exact answer. Let's look at some more examples of estimating decimal differences.

Example 3: Estimate the difference: 36.8 - 5.1

- 36.8- 37

-   5.1-   5
- 36.832

Answer: Rounding to the nearest one, we get an estimated difference of 32.

Example 4: Estimate the difference: 156.871 - 132.15

Analysis: Rounding to the nearest hundredth does not help us to get a rough calculation. For this problem, it is easier to round to the nearest ten.

- 156.871- 156.87
- 132.150- 132.15
- 156.871???.??

Analysis: Let's try rounding to the nearest ten.

- 156.871- 160
- 132.150- 130
- 156.871- 30

Answer: Rounding to the nearest ten, we get an estimated difference of 30.

Example 5: Estimate the difference: 17.54 - 6.39

Analysis: Rounding to the nearest tenth does not help us to get a rough calculation. For this problem, it is easier to round to the nearest one.

17.54   17.5
-   6.39-   6.4
17.54- ??.?

Analysis: Let's try rounding to the nearest one.

17.54-  18
-   6.39-    6
17.54  12

Answer: Rounding to the nearest one, we get an estimated difference of 12.

Example 6: Estimate the difference: 43.9658 - 18.9507

Analysis: Rounding to the nearest thousandth does not help us to get a rough calculation. For this problem, it is easier to round to the nearest tenth.

- 43.9658- 43.966

- 18.9507- 18.951

- 40.9651- ??.???

Analysis: Let's try rounding to the nearest tenth.

- 43.9658- 44.0 = 44

- 18.9507- 19.0 = - 19

- 43.9658- 44.0=- 25

Answer: Rounding to the nearest tenth, we get an estimated difference of 25.

Example 7: Estimate the difference: 6.871 - 2.15

Analysis: Rounding to the nearest hundredth does not help us to get a rough calculation. For this problem, it is easier to round to the nearest one.

- 6.871-16.87
- 2.150-  2.15
17.54-?.??-2

Analysis: Let's try rounding to the nearest one.

- 6.871-17
- 2.150-  2
17.54-52

Answer: Rounding to the nearest one, we get an estimated difference of 5.

In Examples 4 through 7, we estimated using trial and error. If rounding to one place did not work, we tried rounding to another place. In general, it is easier to estimate decimal differences by rounding to the nearest one.

Example 8: Maria has \$4. Will she be able to buy a sandwich for \$1.89, fruit salad for \$0.79, and milk for \$0.89?

Estimate: Rounding each number up to the nearest one (dollar), we get an estimate of \$4.

Answer: Yes: each number was rounded up, resulting in an overestimate of \$4.

Example 9: Mark owes his brother \$13.25. About how much change will he receive from a \$20 bill?

Analysis: Since no place-value was specified, we can round to any place that yields a reasonable estimate.

Estimate 1: Rounding to the nearest one (dollar), Mark will receive about \$7 in change from his brother.

Estimate 2: Rounding to the nearest tenth (dime), Mark will receive about \$6.70 in change from his brother.

Example 10: Refer to the estimates in Example 9 to answer the questions below.

a) Is Estimate 1 an overestimate or an underestimate? Explain your answer.

Answer: Overestimate: \$7 is greater than the actual difference of \$6.75.

b) Is Estimate 2 an overestimate or an underestimate? Explain your answer.

Answer: Underestimate: \$6.70 is less than the actual difference of \$6.75.

Summary: In this lesson, we learned how to estimate decimal differences. When estimating a difference, the estimate should not exceed the original numbers. Estimation can be used to determine if an answer is reasonable. Estimation is an important life skill to have. Without it, a consumer can get short-changed. Sometimes estimation requires a little trial and error.

### Exercises

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.

 1 Estimate the difference of \$9.67 and \$6.19 by rounding to the nearest one (dollar). ANSWER BOX: \$  RESULTS BOX:
 2 Estimate the difference of 2.995 and 1.997 by rounding to the nearest hundredth. ANSWER BOX:   RESULTS BOX:
 3 Estimate the difference of 63.7943 and 24.2581 by rounding to the nearest ten. ANSWER BOX:   RESULTS BOX:

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.

 4. The difference of two decimals is 47.8943. If Robin overestimated the difference, then which of the following estimates did she use? 47.89447.8947.848 RESULTS BOX:
 5. Carolyn paid 15.98 for a DVD. About how much change should she get from a \$50 bill? \$35.02\$34.00\$35.98\$34.98 RESULTS BOX: