Chapter 5: Using Numbers in Sensible Ways
1.
Describe three different ways that you could mentally calculate 16  25.
Answer:
Some possible ways: 4  4  5  5 = 4  5  4  5= 20  20 = 400
(10  25) + (6  25) = 250 + 150 = 400
(16  20) + (16  5) = 320 + 80 = 400
16  100
4 = 4  100 (after dividing 16 by 4) = 400
2.
For each of the following, mentally calculate the exact answer and write it in the blank.
Use number sense. Then write enough to make clear how you thought.
A)
3618 + 2472 – 2618 – 472 = ________
B)
(25  29) + (25  11) = ________
Answer:
A) 3000. Thinking: 3618 – 2618 = 1000. 1000 + 2472 = 3472. 3472 – 472 =
3000.
B) 1000. Thinking: Given = 25  (29 + 11) = 25  40 = 25  4  10 = 100  10.
3.
Give the exact answer mentally: 73.8 + 511.37 + 24 – 73.8. Write how you thought.
Answer:
535.37, taking advantage of the subtraction of 73.8 and the addend 73.8, then
working with the 11 + 24.
4.
Describe how you would mentally compute the exact result in each of the following
without using the standard algorithm: Your description should be concise and include
the exact result.
A) 234 – 119
B) 12% of 150
2
C) 25 
5
Answer:
A) 115 Change to 235 – 120, then work left-to-right.
B) 18 2% of 150 is 3, and 12% of 150 is 6 times as much as that.
C) 10 52  25 will give the same number, and 15 of 25 is 5.
Show how you would mentally compute the exact results:
A) 3000 – 2575
B) 0.75 x 24
C) 24  13 + 24  7
Answer:
A) 425. One way: 2575 + 25 = 2600, plus 400 to get to 3000.
Second way: Add 25 to each to change to 3025 – 2600.
B) 18. 3/4 of 24. 1/4 of 24 is 6. 3  6 = 18.
C) 480. 24  13 + 24  7 = 24  (13 + 7) = 24  20.
5.
6.
Show how you would mentally compute:
A) 0.75  36
B) 34  12 + 34  8
C) 3458 – 1734 – 400 + 1734
Answer:
A) 27, from 3/4 of 36
B) 680, from 34  (12 + 8)
C) 3058, from 3458 – 400 (the other terms give 0)
7.
Give the exact answer by mental calculation. Then write down your thinking. You are
asked for a second way of thinking in parts B and C.
A) 479.38 + 18.9 + 2.4 – 479.38 = ______________
B) 12  125 = ______________ (Give two ways for part B.)
C) 1714 – 897 = ______________ (Give two ways for part C.)
D) 20% of 45 = ______________
Answer:
A) 21.3. Thinking: The first and last give 0, so it is only a matter of finding 18.9
+ 2.4, which = 20 + 13 tenths, or 20 plus 1 and 3 tenths.
B) 1500. One way: 3  4  125 = 3  (4  125). A second way: 3  4  25  5
= 3  (4  25)  5. Students often offer 10  125 = 1250 plus 2  125 = 250,
so that’s why we ask for two ways here.
C) 817. One way: Change to 1717 – 900 by adding 3 to each. Second way
(missing addend): 897 + 3 = 900. Plus 814 more (perhaps in steps). 817.
D) 9. Thinking: 1/5 of 45.
8.
A person can reasonably calculate the exact answer to 1563 – 198 mentally by
A)
counting on his/her fingers.
B)
calculating 1565 – 200.
C)
calculating 1565 – 200 – 2.
D)
calculating 1600 – 200.
E)
This calculation is impossible to do mentally.
Answer:
B
A person who is calculating the exact answer to 18  15 mentally starts by calculating
2  15. The person would finish the mental calculation by calculating...
A)
9  30
B)
27  10
C)
36  7.5
D)
60  4.5
E)
None of A–D
Answer:
A
9.
10.
For each, describe two different strategies for performing the following computation
mentally.
A) 29 + 58
B) 74 – 28
C) 8  15
Answer:
A) 30 + 57 = 87; 20 + 50 is 70, plus 9 is 79, plus 1 is 80, plus 7 is 87
B) 76 – 30 = 46; 74 – 24 is 50, – 4 more is 46
C) 4  30 = 120; 8  10 + 8  5 = 80 + 40 = 120; 8  5 = 40, 40  3 = 120
11.
Determine the following mentally, writing enough to make your mental work clear.
A) 40% of 80
B) 15% of 300
C) 20% of 14
D) 100% of 71
E) 5% of 60
F) 120% of 20
G) 15 is 25% of ?
H) 14 is 50% of ?
Answer:
A) 32
B) 45
C) 2.8
D) 71
E) 3
F) 24
G) 60
H) 28
12.
Tell how one might mentally compute the following:
A) 25  104
B) 25% of 104
C) 200% of 104
Answer:
Each can be done in a variety of ways. Here are some possibilities:
A) 25  100 + 25 x 4 is 2500 + 100 is 2600 or 100
4 x 104 = 100 x 26 = 2600
B)
1
4
of 104 is
1
4
of 100 +
1
4
of 4 is 25 + 1 is 26.
C) Twice 104 is 208.
NOTE: Only a few ways are shown to estimate a calculation in the problems that
follow. Other ways may also be correct. Calculators should not be allowed for items
involving either estimation or mental computation.
13.
Show how you would estimate:
A) 391  612
B) 0.74  798
C) 32% of 19
D) 196% of 25
Answer:
A) Round 391 to 400 and 612 to 600; 400  600 = 240,000.
B) Around 3/4 of 800, which is 600.
C) 6 is a good estimate because a third of 18 is 6.
D) Just under 50 because 2  25. (Also, since it is 4% of 25, or 1, less than that,
the exact answer, 49, is fairly easy.)
14.
Using benchmarks, find an estimate of each of the following and explain your
reasoning.
A) 7.5 % of $594.
B) 0.32147  (67.557% of 89.4853)
C) 2 89  ( 13
24  0.99184)
D) 5.8 ÷ 12
Answer:
A) 45. 7.5% is 3/4 of 10%; 10% of 600 is 60, and 3/4 of $60 is $45.
B) 20. The estimate should be about 1/3 of 2/3 of 90, or about 1/3 of 60.
(Students may also offer 1/3 of 2/3 of 90, or 2/9 of 90.)
C) About 3 1/2. Almost 3, plus about 1/2 of 1
D) About 1/2. 6 ÷ 12 is 1/2.
Joe Blue was estimating 92  31. He said that rounding 31 to 30 then taking 92  30
(which is 2760) is a better estimate than rounding 92 to 90 and then taking 90  31
(which is 2790) because in the first case you lost only 1 by rounding 31 to 30, but in
the second case you lost 2 by rounding 92 to 90. Explain how Joe's reasoning is
incorrect. Your answer should show number sense.
Answer:
Estimating 92  31 via 92  30 will be “off” by 92. Estimating 92  31 via 90 
31 will be “off” by only 2  31, or 62.
15.
16.
For each of the following, mentally obtain an estimate of the answer and write it in the
blank. Use number sense. Then write enough to make clear how you thought.
A) 34% discount on an $89 suitcase. _________
B) 0.26  43,135 ≈ ___________
C) 61  334 ≈ ______________
D) 74.35%  1195  0.9837 ≈ ______________
E) (1201.794 ÷ 0.25) + 0.0423 ≈ ______________
Answer:
A) About $30. Thinking: 34% is about 1/3, and $89 is about $90. 1/3 of $90 is
$30.
B) About 11,000. Thinking: 0.26 is about 1/4; round 43,135 to 44,000.
C) About 20 000. Thinking: 60  1/3 of 1000 = 20 000.
D) About 900. Thinking: 3/4  1200  1.
E) About 4800. Thinking: 1200 ÷ 1/4. Definitely take off points for answers
such as 4800.0423.)
17.
Is 40 ÷ 1.99 less than, equal to, or greater than 20? Explain, showing your
understanding of a meaning of division.
Answer:
Greater than 20 since there will be more 1.99s in 40 than there are 2s.
18.
Why is this student not showing good number sense in estimating 249.738 + 48.246?
“Well, 249 is about 250, and 48 is about 50. 250 + 50 is 300. Then .738 + .246 is about
1. So, my estimate is 301.”
Answer:
In using 250 and 50 for the whole number parts, the student is already making an
over-estimate. Working with the decimals would only make the estimate farther
off.
19.
0.7614987  159.23842 is about
A)
1.2.
B)
12.
C)
120.
D)
1200.
E)
None of A–D
Answer:
C (from ¾ of 160)
20.
1.334496  301.66 is closest to
A)
400.
B)
391.
C)
390.
D)
40.
E)
39.
Answer:
A
21.
278.4132987 x 0.2617944285 is about
A)
7.
B)
70.
C)
700.
D)
7000.
E)
None of A–D
Answer:
B
22.
0.3347876  629.847291143 is closest to
A)
1800.
B)
180.
C)
200.
D)
210.
E)
240.
Answer:
D
23.
Use number sense in locating the decimal point in the answer. Explain your thinking
briefly.
A) 77.5  2.84 = 2 2 0 7
B) 1002.6 ÷ 3.6 = 2 7 8 5
Answer:
A) 220.7. Explanation: The product is about 80  3 = 240.
B) 278.5. Explanation: The quotient is about 1000 ÷ 4 = 250.
24.
You are using a calculator that can show only eight digits of an answer. When you
finish, you realize that you forgot to press the decimal points in the original numbers.
Show where the decimal points go and explain how you know, using number sense (no
paper-pencil calculation).
A) 9.87429  637.21945 = 6 2 9 2 0 8 9 6
B) 413.69824 ÷ 24.92617 = 1 6 5 9 6 9 4 4
Answer:
A) 6292.0896. Thinking: The product should be around 10  630 = 6300.
B) 16.596944. Thinking: The quotient should be around 400 ÷ 25, or 16 (how
many 25s make 400).
25.
Demonstrate number sense in estimating the following expressions and responding to
the question about your answer in part A. Write enough to make your thinking clear.
A) 75.48%  883.375  567
566 is about ____________.
Is your estimate (less than, equal to, greater than) the exact answer? (Don't figure the
exact answer out!) ______________ because
B) 32% of $595.45
C) 60% of $271
D) 87  52
Answer:
A) About 660: 3/4  880  1. Since each of the new factors is less than the
original, 660 will be less than the actual answer (Aside—from a calculator:
The exact answer is a bit more than 667.949).
B) About 1/3 of $600 is $200.
C) $150: 3/5 of $250, or $180 from 2/3 of 270.
D) 4500 from 90  50.
26.
Company X bought 17,569 truckloads worth $24,598 each. Company Y bought 24,598
truckloads worth $17,569 each. Which company paid more, and how do you know?
Answer:
The companies paid the same. 17569  24598 = 24598  17569.
27.
Betty was asked to give her best estimate for 25% of 7991.8.
She estimates by taking 14 of 8000, which is 2000, and taking 14 of 0.8, which is 0.2.
Therefore, her estimate is 2000.2
Comment on Betty’s reasoning.
Answer:
Betty does not seem to realize that in using 8000 for 7991.8, the 0.8 is already
involved. (She likely used the 8000 for just the 7991.) Using 8000 introduces
an error that would only be exaggerated by adding a number-senseless 1/4 of
0.8. Another point that may not have come up in class is that reporting an
estimate with tenths, like 2000.2, implies that the estimate is quite close, down to
the nearest tenth. That clearly is not the case here.
28.
42,189 ÷ 511,264 is about how many percent?
A)
8%
B)
12.5%
C)
80%
D)
125%
E)
None of A–D
Answer:
A from 40/500 = 4/50 = 8/100 = 8% , or possibly 40 ÷ 500 = (40 ÷ 5) ÷ 100 = 8
÷ 100 = 8%
29.
Estimate the following with a brief explanation:
A) 65% of 37
B) 140% of 52
C) 18% of 971
D) 89 % of 120 = ___ (decimal)
Answer:
A) 24 from 2/3 of 36
B) 70 from 7/5 of 50
C) 195 from 1/5 of 975 (200 probably okay, from 1/5 of 1000)
D) 1.2 from 0.01 from 8/9 being close to 1, a number that is easy to calculate
with
30.
Use benchmarks to estimate the following. Explain how you estimated.
A) 60% of $271
B) 87  52
C) 32% of $595.45
Answer:
A) $150 3/5 of $250, or $180 from 2/3 of 270, or $162 from 50% of $270, plus
10% of $270
B) 4500 from 90  50
C) About $200, 1/3 of $600
31.
Estimate the indicated quantity and tell how you did it.
A) 15% of $51.07
B) 25% of 1998
C) 125% of 47
D) 48% of 212
E) 4% of 201
Answer:
A) 10% of 50 is 5, and 5% of 50 is 2.5, so the answer is about 7.5.
B) 14 of 2000 is 500.
C) 100% of 47 is 47;
1
4
of 48 is 12, so the answer is about 59.
D) Half of 212 is 106 (or half of 200 is 100).
E) 4  200 is 800, so 8; or 1% of 200 is 2, so 4% is 4  2 is 8.
32.
Approximately how long is this line segment, in inches or in centimeters? Be sure to
name the measuring unit you use.
_________________________________________________________
What benchmark did you use?
Answer:
Measure this line segment before you give it to students.
33.
Name an item that costs approximately $500,000.
A)
A family car
B)
A jet plane
C)
A nice home
D)
A stereo
Answer:
C
34.
Name two types of estimates you use in daily life.
Answer:
Could be length of time to drive to campus, amount of money to fill tank with
gas, amount of money needed at cash register, cost of meals for a week, etc.
35.
Using benchmarks, find an estimate of the time it would take, averaging 50 mph for 8
hours a day, to drive across the United States.
Answer:
Estimating that the distance across the United States is about 3000 miles, the trip
would take 3000 ÷ 50 = 60 hours, or 60 ÷ 8 = 7 1/2 driving days of 8 hours each.
36.
Write in scientific notation.
A) 38,000,000,000
B) 382.45
10
2
Answer:
A) 3.8  10
B) 3.8245  10
C) 0.000 000 000 456
-10
C) 4.56  10
Write the following values in scientific notation.
A) The earth is 150,000,000,000 meters from the sun.
B) The speed of light is 300,000,000 meters per second.
C) A dust particle is 0.000 000 000 753 kg.
11
8
-10
Answer:
A) 1.5  10
B) 3  10
C) 7.53  10
37.
38.
How many seconds does it take light to reach the earth from the sun? Express your
answer in scientific notation. (See item 37 for relevant data.)
11
8
10
8
2
Answer:
(1.5  10 ) ÷ (3  10 ) = (15  10 ) ÷ (3  10 ) = 5  10 or 500 seconds; a
little over 8 minutes.