UNIT ONE: Prealgebra in a Technical World
SWBAT 1.
Estimate to determine if a result is probably correct.
2.
Round down to estimate results.
3.
Round up to estimate results.
4.
Round to the nearest to estimate calculated results.
5.
Determine whether a calculator is truncating or not.
Cheryl thinks she entered 23 ∙ 45 on her calculator. The calculator screen reads “ 92 .”
How can she tell if this answer is correct or incorrect?
Cheryl has a job to do and she has to do that job quickly, but if her answer is terribly wrong, she could lose money. So Cheryl rounds and makes an estimate to check her calculator result.
In this section we study rounding numbers and making estimates to check results.
Cheryl estimates by multiplying 20 ∙ 50 in her head. She knows this product is 1000.
Using her estimate, she knows her calculator has the wrong answer and simply keys in the problem, 23 ∙ 45 , again. The second time the result is 1035. This answer checks with Cheryl’s estimate.
Like Cheryl, to be successful using technology, we do not use paper and pencil to check calculators. We need to use math facts and good estimating skills to check these results.
Cheryl’s original answer was so far away from her reliable estimate that we say her calculator’s answer was “out of the ballpark.” What we mean is that the first calculator result
cannot be correct. We need to throw out such results and calculate again.
If, instead, the calculator result is close to a reliable estimation, then it is probably
correct. We do not know if the answer is exactly correct, but often just knowing that we are “in the ball park” with a result is enough. Even when we have to check our answer further, knowing if we are probably correct is the best first step.
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4 SECTION 1.1 Estimating Results by Rounding
People make estimates by rounding in different ways. You may complete the problems below differently than your author, but if you round numbers and do the math in your head, you are estimating correctly.
Example 1: Midge and Steve are computing their salaries. They each make $9.54 per hour and will work 40 hours this week. Both use a calculator. Midge’s result is $381.60. Steve’s result is
$781.60. Whose result is probably correct and whose cannot be correct?
Think it through: Because $9.54 ∙ 40 is about $10 ∙ 40, a good estimate is $400.
ANSWER: Midge’s answer is probably correct. Steve’s answer is far too high.
Example 2: Sharon has a new job paying $10.23 per hour. She will be working 19 hours per week. Before taxes, about how much will she earn in a week?
Think it through: Round 10.23 to 10 and 19 to 20, 10 ∙ 20 = 200.
ANSWER: Sharon will earn about $200 per week before taxes.
Example 3: Brian is charged $29.49 when buying “Smiley Meals” that cost $3.69 each for 7 children. Brian lives in Oregon, so he pays no sales tax. Estimate the value to determine whether Brian was charged correctly.
Think it through: $3.69 is less than $4, so a high estimate would be $4 ∙ 7 = 28.
ANSWER: Brian was overcharged.
Example 4: About how much will you pay for your phone in one year (12 months) if your plan is
$57 per month?
Think it through: Round the months to 10 and the payment to 60, 10 ∙ 60 = 600
ANSWER: You will pay about $600 for your phone in one year.

UNIT ONE: Prealgebra in a Technical World
(Notice that 12 x 60 gives a much closer estimate. You are always free to use more digits.
Knowing your 11 and 12 multiplication facts would make this estimate far more accurate, since these two numbers show up often in real world problems.)
After rounding in the examples above, we could “do the math in our heads;" that is we could use mental math. Notice that we had to round first. Rounding is the process of approximating a number by choosing one that is close to the original, but which is easier to work with.
Even when you are not estimating, mental math is the easiest way to calculate. All
mental math depends on your memorization of math facts.
STUDY SKILLS: Now is a good time to make sure you can add, subtract, and multiply single digits mentally. In addition, you must recognize your division facts.
Your instructor has extra resources that you can use to learn the math!
Most of us have already learned to “round to the nearest” but, outside of math classrooms, we often “round up” or “round down” so that our estimate is “at least enough” or
“not too much.” For instance, I might round down so that I do not give a 5-pound dog or a young child too much medication. But I will definitely round up all my bills so that I have enough money at the end of the month.
No matter whether we round up, down, or to the nearest, the quickest way to estimate is to round to the leading digit. The leading digit is the furthest left non-zero digit of a
number, the first number we read.
For example: 5,400
“5” is the leading digit.
$6.42
“6” is the leading digit.
0.00392
“3” is the leading digit.
Rounding to the leading digit gives an accurate approximation of the number while always leaving a number that we can calculate with easily.
5

6 SECTION 1.1 Estimating Results by Rounding
Round DOWN to Estimate
We round down to make estimates when we want to make a low estimate. We round down to the leading digit using these steps:
1.
Keep the leading digit in your number.
2.
Replace all digits to the right with zeros.
Number
Round down to the leading digit
47,000
40,000
34.9725
30
0.0098
0.009
990
900
5,000
5,000
5,999
5,000
4
4
Once you have rounded your numbers, you are ready to use mental math to calculate.
Example 5: Chad is leaving a job interview. He has been offered a job that pays $726 per week.
This job pays him for all 52 weeks out of the year and includes holidays. He wants to know if this salary is high enough. What is a good estimate for the proposed annual salary?
Think it through: Chad rounds both numbers down so his estimate will be at least enough.
700 ∙ 50 = 35,000.
ANSWER: Chad will make at least $35,000 per year before taxes at this job.
Round UP to Estimate
We round up to make estimates when we want to make a high estimate. If the leading digit of the number is the only non-zero digit, then the number is already rounded up. For instance, 3,000 is rounded up. We round up to the leading digit using these steps:
1.
If the leading digit in your number is followed by any digits other than zero, in any place value, add 1 to the leading digit.
2.
Replace all digits to the right with zeros.
Number
Round up to the leading digit
47,000 34.9725 0.0098 990
50,000 40 0.01 1,000
5,000 5,999
5,000 6000
4
4

UNIT ONE: Prealgebra in a Technical World
Example 6: At the grocery store Sam fills his cart with items that cost: $2.79, $1.49, $6.49,
$2.69, $3.79, $0.95, $4.29, $1.69, $2.89. About how much money will he need to hand the cashier?
Think it through: Round up so we have enough money: 3 + 2 + 7 + 3 + 4 + 1 + 5 + 2 + 3 = 30
ANSWER: If Sam has at least $30, he knows he has enough money to pay for these groceries.
Round to the NEAREST to Estimate
Rounding to the nearest is the acceptable method of rounding for entering business transactions and scientific data.
To round to the nearest leading digit, we take these steps:
1. Look at the second digit, the digit immediately to the right of your leading digit.
2. If the second digit is 5 or more, round the leading digit up.
3. If the second digit is less than 5, round the leading digit down.
Number 47,000 34.9725 0.0098 990 5,499 5,500 4
Round to the nearest leading digit
50,000 30 0.010 1,000 5,000 6,000 4
Example 7: Ivan is planning a trip to Alaska. His car gets 28 miles per gallon and, using
Mapquest.com, he finds that his trip will be 5,867 miles. What is a good estimate for the number of gallons he will need for his trip?
Think it through: Ivan rounds the miles to the leading digit: 6,000. He rounds his gas mileage to the leading digit: 30. He calculates: 6000 ÷ 30 = 200.
ANSWER: Ivan will buy about 200 gallons of gas.
7

8 SECTION 1.1 Estimating Results by Rounding
In many applications we sometimes round to a specific place value.
Example 8: The California Department of Corporations gives these instructions for entering numbers on an annual report form:
“All items requesting dollar amounts are to be rounded to the nearest whole dollar. Do not add ".00" to represent rounding to the nearest dollar. Do not round items to the nearest
thousand or million dollars.” 1
How would these numbers be reported? a.
2.50 b.
$3,875,489.96 c.
$43,000.47
Think it through: Round to the nearest dollar and do not include “.00.”
ANSWER: a. $3 b. $3,875,490 c. $43,000
To round to the nearest, look only at the digit just to the right of the place you
are rounding. On the number line below, 65 is the middle point between 60 and 70. No matter how many digits we have in a number less than 65, any number below 65 is still closer to 60. In this case 64.9999 is closer to 60, not
70. Do not be confused by how many digits you see in a number; find where the number is on the number line:
Example 9: Jim, a real estate appraiser, rounds each item to the nearest hundred and then adds the values together when writing appraisals. He rounds this sum to the nearest thousand dollars for his final appraised value. Jim is appraising a property. Before rounding, the driveway is valued at $6,876, the septic system at $4,549, the acreage at $83,739 and the house at $102,153. What is the total appraised value for this property?
1
California Department of Corporations. (2008). California deferred deposit transaction activity report for the year ended 2008. Retrieved July
28, 2009, from: http://www.corp.ca.gov/forms/pdf/2030.doc

UNIT ONE: Prealgebra in a Technical World
Think it through: Jim first rounds each value to the nearest hundred dollars and adds:
6,900 + 4,500 + 83,700 + 102,200 = 197,300. Jim then rounds to the nearest thousand dollars.
ANSWER: Jim appraises the value of the property at $197,000.

Check Point 1 a.
You have agreed to babysit for 21 days at $22 each day. Find a low estimate of your total pay. b.
The most expensive wood in the world is agarwood, a tropical wood that has been infected with mold. The infection results in an aromatic resin that is refined and sold to make perfume. Currently 18 grams of agarwood is advertised for $158. About how much is agarwood per gram? c.
Like many Americans, Justin is overweight. His doctor tells him to lose 54 lbs in one year.
Tell Justin about how many pounds he needs to lose each month to reach his goal.
Truncate & Calculate
Another name for rounding down is truncating. Truncate means to cut off. When we round down, we cut off digits, leaving only zeros for place holders. Some calculators, at the very end of their displays, truncate. Other calculators round to the nearest. It is good to know just how any calculator you use is rounding numbers.
Example 10: To determine whether their calculators are truncating, Annette divides 2 by 3 and sees “0.666666666.” Abigail, on the other hand saw “0.666666667” when she used her calculator to divide. Which calculator is truncating and which is rounding?
9

10 SECTION 1.1 Estimating Results by Rounding
Think it through: Both calculators have the correct rounded number, but they are using different methods. Since 6 rounds up to 7 and down to 6, we know which way each calculator is rounding.
ANSWER: Annette’s calculator is truncating. Abigail’s calculator is rounding to the nearest.

Check Point 2
Check all calculators in your home. Below write down how many calculators are
Rounding to the nearest ______ Truncating ______
And bring these results to class.

UNIT ONE: Prealgebra in a Technical World
Name _______________________________
For problems 1 through 4, round each number to the indicated place value.
1.
Round up to the leading digit
Round down to the leading digit
Round to the nearest ten
Round up to the hundred
361 849 908 979 8,532 654
2.
Round to the nearest leading digit
Round to the nearest hundred
Round down to the leading digit
Round down to the ten
545
3.
249 140 5,040 222 9,902
Round up to the dollar
$3.61 $8.49 $9.08 $9.79 $85.32 $6.54
Round down to the dollar
Round to the nearest dime
4.
Round to the nearest dollar
$5.45 $2.49 $1.40 $50.40 $2.22 $99.02
Round to the nearest leading digit
Round down to the nearest dime
For problems 5 to 12, circle the best estimate for the following.
5. 78 + 29

7. 51 – 13 
9. 19 + 87 
11. 532 + 97 
110
30
100
640
130
20
110
630
120
40
120
620
6. 95 + 64 
8. 173 – 52
10. 25 + 789
12. 862 + 77



130
120
910
940
180
170
820
930
160
70
810
920
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12 SECTION 1.1 Estimating Results by Rounding
Besides checking correctness of answers using the leading digit, the last digit of an answer provides a check for correctness. If the last digit of an answer is not correct, the answer is “not correct.” For problems 13 to 16 check the appropriate box. (The first problem is done for you.)
13. 160 + 152 = 260
14. 391 - 247 = 144
15. 187
•
138 = 25,806
16. 154 • 276 = 27,501
Last Digit
Correct
Last Digit
Not Correct

For problems 17 to 28 write the rounding you do to estimate, and then circle PC for “probably correct” or NC for “not correct.” (The first problem is done for you.)
Problem
17. 380 • 48 =18,240
18. 700 • 770 =539,000
19. 16 + 60 + 32 + 77 = 315
20. 506 – 46 – 83 = 377
21. 49 + 81 + 67 + 76 = 223
22. 662 – 290 – 16 = 356
23. 809 – 350 – 21 = 278
24. 70 + 77 + 30 = 207
25. 220 • 323 = 5430
26) 185 • 98 = 18,130
Estimate
400 x 50 = 20,000 PC NC
PC NC
PC NC
PC NC
PC NC
PC NC
PC NC
PC NC
PC NC
PC NC

UNIT ONE: Prealgebra in a Technical World
For problems 27 and 28, write the answer only.
27. An item is on sale at 4 for $1.59.
If you buy just one, how much will the store charge?
28 . If soap is priced at 4 for $2.99.
How much will you pay for 2 bars?
For problems 29 to 38, write your calculations using rounded numbers. Write and circle your estimate. (Sometimes your estimate may be exactly correct.)
29. Crunchy Toffee cookies come in packages of one dozen. If you have 18 children at a party and you want to make sure they
30. Baseboard comes in 16- foot lengths.
You need to buy 470 feet of baseboard.
How many pieces of baseboard will you need to buy? have 6 cookies each, about how many packages will you have to buy?
31. Potting soil comes in 2-cubic feet bags.
You wish to fill a deck planter with
33.
43 cubic feet of potting soil. About how many bags will you need to buy?
You are going to construct a frame for your friend’s odd shaped painting. About how much molding should you buy if the art measures 27 cm, 37 cm, 57 cm and
58 cm?
35. Your shopping cart has items that cost
$2.99, $0.89, $3.79, and $3.79. You realize that you forgot your debit card, and you only have $12 in your pocket. Do you have enough money to pay for items in your cart?
32. How much will you spend for gas on your trip if your car averages 15 mpg, your trip is 1,300 miles, and gasoline costs
$2.89 per gallon?
34. One gallon of paint will cover approximately 390 square feet of wall space. If your walls total 1,470 square feet and you have to subtract away 440 square feet for windows and doors, how many gallons will you need to buy?
36. Your shopping cart has items that cost
$1.99, $3.99, $2.89, $3.75, and $3.25.
You realize that you forgot your debit card and you only have $15. Do you have enough money to buy the items in your cart?
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14 SECTION 1.1 Estimating Results by Rounding
37. Your monthly bills are $370, $55.19,
$34.59, $39.59 and $26.49. If you want to have at least $80 for entertainment, will your paycheck of $699 be enough?
(Remember to use estimation—it is easier!)
38. Your bills for the month are $700,
$198.22, $55.42, $148.37, and $44.05. If you want to have at least $160 for entertainment, will your paycheck of
$1,337 be enough? (Remember to use estimation—it is easier!)
Knowing your math facts and using paper and pencil skills for simple operations is even more important now that we use technology. If you find that you have difficulty completing any of the problems below, your teacher has information and materials for you to help you remember this math.
39.
328 40.
1,836 x 35 x 387
41.
39
)
986 42.
76
)
2,384
45.
20 ) 748 46.
76 ) 2,000 43.
907 44.
7,777 x 14 x 309