Deck 5: Using Numbers in Sensible Ways

Question 1

Which calculation does NOT show how to mentally calculate 16 × 25?

A) 4 × 4 × 5 × 5 = 4 × 5 × 4 × 5 = 20 × 20 = 400
B) (10 × 25) + (6 × 25) = 250 + 150 = 400
C) (16 × 20) + (16 × 5) = 320 + 80 = 400
D) 4 × 100 = 400
E) This calculation is impossible to do mentally.

Answer 1

4 × 100 = 400

Question 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 your thought process clear.

Answer 2

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.

Question 3

Give the exact answer mentally: 73.8 + 511.37 + 24 - 73.8. Describe your thought process.

Answer 3

535.37, taking advantage of the subtraction of 73.8 and the addend 73.8, then working with the 11 + 24

Question 4

Describe how you would mentally compute the exact result in each calculation without using the standard algorithm. Your description should be concise and include the exact result.

A) 234 - 119
B) 12% of 150
C) 25 × 2/5

Question 5

Show how you would mentally compute the exact results of each calculation.

A) 3000 - 2575
B) 0.75 × 24
C) 24 × 13 + 24 × 7

Question 6

Show how you would mentally compute each calculation.

A) 0.75 × 36
B) 34 × 12 + 34 × 8
C) 3458 - 1734 - 400 + 1734

Question 7

Give the exact answer by mental calculation, and 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 = _____

Question 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.

Question 9

A person who is calculating the exact answer to 18 × 15 mentally starts by thinking about factors of 18. The person would finish the mental calculation by calculating:

A) 9 × 30.
B) 27 × 10.
C) 36 × 7.5.
D) 60 × 4.5.

Question 10

For each, describe two different strategies for performing the following computation mentally.

A) 29 + 58
B) 74 - 28
C) 8 × 15

Question 11

Determine each calculation 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 ?

Question 12

Tell how one might mentally compute each calculation.

A) 25 × 104
B) 25% of 104
C) 200% of 104

Question 13

Show how you would estimate:

A) 391 × 612.
B) 0.74 × 798.
C) 32% of 19.
D) 196% of 25.

Question 14

Using benchmarks, find an estimate of each calculation and explain your reasoning.

A) 7.5% of $594
B) 0.32147 × (67.557% of 89.4853)
C)
D) 5.8 ÷ 12

Question 15

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.

Question 16

For each calculation, mentally obtain an estimate of the answer and write it in the blank. Use number sense. Then write enough to make your thought process clear.

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 ?_____

Question 17

Is 40 ÷ 1.99 less than, equal to, or greater than 20? Explain, showing your understanding of a meaning of division.

Question 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."

Question 19

0.7614987 × 159.23842 is about:

A) 1.2.
B) 12.
C) 120.
D) 1200.
E) 12,000.

Question 20

1.334496 × 301.66 is CLOSEST to:

A) 400.
B) 391.
C) 390.
D) 40.
E) 39.

Question 21

278.4132987 × 0.2617944285 is about:

A) 7.
B) 70.
C) 700.
D) 7000.
E) 70,000.

Question 22

0.3347876 × 629.847291143 is CLOSEST to:

A) 1800.
B) 180.
C) 200.
D) 210.
E) 240.

Question 23

Use number sense in locating the decimal point in the answer. Explain your thinking briefly.

A) 77.5 × 2.84 = 2 2 0 1
B) 1002.6 ÷ 3.6 = 2 7 8 5

Question 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

Question 25

Demonstrate number sense in estimating the following expressions and responding to the question about your answer. Write enough to make your thinking clear.
75.48% × 883.375 × 567/566 is about _____.
Is your estimate less than, equal to, greater than the exact answer? Why? (Don't figure the exact answer out!) _____

Question 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?

A) The companies paid the same.
B) Company X paid more.
C) Company Y paid more.

Question 27

Betty was asked to give her best estimate for 25% of 7991.8. She estimates by taking 1/4 of 8000, which is 2000, and taking 1/4 of 0.8, which is 0.2. Therefore, her estimate is 2000.2. Comment on Betty's reasoning.

Question 28

42,189 ÷ 511,264 is about how many percent?

A) 8%
B) 12.5%
C) 80%
D) 125%
E) 1.25%

Question 29

Estimate each calculation with a brief explanation.

A) 65% of 37
B) 140% of 52
C) 18% of 971
D) 8/9% of 120 = _____ (decimal)

Question 30

Use benchmarks to estimate each calculation. Explain how you arrived at the estimate.

A) 60% of $271
B) 87 × 52
C) 32% of $595.45

Question 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

Question 32

Name an item that costs approximately $500,000.

A) a family car
B) a private jet
C) a nice home
D) a stereo

Question 33

Name two types of estimates you use in daily life.

Question 34

Using benchmarks, what is the BEST estimate of the time it would take, averaging 50 mph for 8 hours a day, to drive across the United States (about 3000 miles)?

A) 6.25 driving days
B) 7 $\frac { 1 } { 2 }$ driving days
C) 400 driving days

Question 35

Write in scientific notation.

A) 38,000,000,000
B) 382.45
C) 0.000 000 000 456

Question 36

Write each value 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.

Question 37

How many seconds does it take light to reach the Earth from the sun? Express your answer in scientific notation. (The Earth is 150,000,000,000 meters from the sun, and the speed of light is 300,000,000 meters per second.)

Question 38

Kyle estimated that 23 × 31 was 603. Use a rectangular area model to illustrate the misunderstanding and the correct product.