Deck 5: The Nature of Numbers

Question 1

Solve the equation for x. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Answer 1

​

Question 2

Perform the operation using arithmetic for a 12-hour clock. ​

A)
B)
C)
D)
E) does not exist

Answer 2

Question 3

Assume that today is Monday (day 2). Determine the day of the week it will be at the end of the period at 2 years. (Assume no leap years.) ​

A) Friday
B) Tuesday
C) Sunday
D) Wednesday
E) Thursday

Answer 3

Wednesday

Question 4

Is the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} a group for multiplication modulo 11? ​
​

A) No
B) Yes

Question 5

Solve the equation for x. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 6

Perform the operation.
​
​

Question 7

Suppose you make six round trips to visit a sick aunt and wish to record your mileage to the nearest mile. You forget the original odometer reading, but you do remember that the units digit has increased by 8 miles. If you know that your aunt lives somewhere between 26 and 31 miles from your house, how far exactly is her house? ​

A) 29 mi
B) 28 mi
C) 31 mi
D) 27 mi
E) 26 mi

Question 8

Perform the operation. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 9

Perform the operation using arithmetic for a 12-hour clock. ​

A)
B)
C)
D)
E)

Question 10

An International Standard Book Number (ISBN) is used to identify books. The ISBN number for some book is 0-534-36890-5. The first digit, 0, indicates the book is published in an English-speaking country. The next three digits, 534, identify the publisher (Brooks/Cole) and the next five digits identify the particular book. The last digit is a check digit, which is used as follows. Multiply the first digit by 10, the next digit by 9, and so on: ​
10(0) + 9(5) + 8(3) + 7(4) + 6(3) + 5(6) + 4(8) + 3(9) + 2(0) + 1(5) = 209
​
This number is congruent to 0, (mod 11). This is not by chance. The check digit, 5, is chosen so that this sum is 0, (mod 11). In other words, the check digit x for the book with ISBN number 0-534-34015-x is found by considering:
​
10(0) + 9(5) + 8(3) + 7(4) + 6(3) + 5(4) + 4(0) + 3(1) + 2(5) = 148 ≡ 5
​
This means that for check digit x, 5 + x ≡ 0, (mod 11). The one-digit solution to this equation is x = 6, so the check digit is 6. What is the check digit for the book with ISBN 0-8028-1430-x?
​

A)
B) ​
C) ​
D) ​
E) ​

Question 11

Perform the indicated operation. ​ ​

A) 4
B) 3
C) -2
D) 2
E) 0

Question 12

Is the statement true?
​

Question 13

Perform the operation using arithmetic for a 12-hour clock.

Question 14

Perform the operation using arithmetic for a 12-hour clock.

Question 15

Perform the indicated operations.
​ ​ __________
​ __________

Question 16

Solve the equation for x.
​ ​
Assume that . If there is no solution, answer none. If you get several answers, enter one of them.

Question 17

Suppose you make six round trips to visit a sick aunt and wish to record your mileage to the nearest mile. You forget the original odometer reading, but you do remember that the units digit has increased by 8 miles. If you know that your aunt lives somewhere between 2 and 5 miles from your house, how far exactly is her house?
​
__________ miles

Question 18

An International Standard Book Number (ISBN) is used to identify books. The ISBN number for some book is . The first digit, 0, indicates the book is published in an English-speaking country. The next three digits, 534, identify the publisher (Brooks/Cole) and the next five digits identify the particular book. The last digit is a check digit, which is used as follows. Multiply the first digit by 10, the next digit by 9, and so on:
​
​ ​
This number is congruent to 0, (mod 11). This is not by chance. The check digit, 5, is chosen so that this sum is 0, (mod 11). In other words, the check digit x for the book with ISBN number is found by considering:
​ ​
This means that for check digit x, . The one-digit solution to this equation is , so the check digit is 6. What is the check digit for the book with ISBN ?
​
x = __________

Question 19

Is the statement true or false?

Question 20

Assume that today is Monday (day 2). Determine the day of the week it will be at the end of the period at 155 days. (Assume no leap years.)

Question 21

Carry out the operations with decimal forms. ​
​ ​

A) 1,699.44
B) 41
C) 408
D) 94.81
E) 184.9

Question 22

Check whether this set and operation form a group. ​ ​

A) no
B) yes
C) in definably

Question 23

Express the number as a decimal. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 24

Check whether the set and the operation form a group. ​
Ω for ×
​

A) no
B) yes

Question 25

Express the number as a decimal. Give your answer to two decimal places, if required.
​

Question 26

Let be an arbitrary operation. Describe the operation for the problem. ​ ​

A) a + b + 1 - 1
B) a + b + 1
C) a + b
D) ab
E) a - b

Question 27

Carry out the operations with decimal forms. Give your answer to three decimal places.
​

Question 28

Change the terminating decimal to fractional form. ​
6)98
​

A)
B) ​
C) ​
D) ​
E) ​

Question 29

Identify the illustrated property. ​
​ ​

A) identity property
B) inverse property
C) distributive property
D) associative property
E) commutative property

Question 30

Express the number as a decimal. ​ ​

A)
B)
C)
D)
E) ​

Question 31

Express the number as a decimal.
​ ​
Use an overbar to indicate the numerals that repeat, if needed.

Question 32

Identify the illustrated property as commutative, associative, distributive, or inverse.
​

Question 33

Let be an arbitrary operation. Describe the operation for the problem. ​ ​

A) ab + 1
B) a + b
C) ab
D) 2a - b
E) a + b + 1

Question 34

Let be an arbitrary operation.
​
​ ​
Describe the operation .

Question 35

Is the set {0, 1, 2, 3, 4, 5, 6} a group for addition modulo 7? Answer yes or no.

Question 36

Carry out the operations with decimal forms. Give your answer to one decimal place, if required.
​

Question 37

Find a rational number and an irrational number between the given pair of numbers. ​
​4)5 and 4.6
​

A)
B) ​
C) ​
D) ​
E) ​

Question 38

Carry out the operations with decimal forms. ​ ​

A) 1.637
B) - 1.637
C) 6.013
D) - 6.013
E) 2.637

Question 39

Change the terminating decimal to fractional form.
​
0.63

Question 40

Let be an arbitrary operation.
​ ​
Describe the operation .

Question 41

Check whether this set and operation form a group. Answer yes, no, or in definably.
​

Question 42

Simplify the expression. ​ ​

A) ​
​
B) ​
C) ​
D) ​
E) ​

Question 43

How far from the base of a building must a 13-ft ladder be placed so that it reaches 5 ft up the wall? ​

A) 18 ft
B) 12 ft
C) 13 ft
D) 8 ft
E) 5 ft

Question 44

How far from the base of a building must a 26-ft ladder be placed so that it reaches 10 ft up the wall?
​
__________ ft

Question 45

Find an irrational number between 4 and 8. ​

A) 3.232332333....
B) 9.454554555....
C) 4.222....
D) 4.234234234....
E) 4.232332333....

Question 46

Use the definition of square root to find the indicated product.
​

Question 47

Check whether the set and the operation form a group. Answer yes or no.
​
​Ω for +

Question 48

Find a rational number and an irrational number between the given pair of numbers.
​
4.5 and 4.6

Question 49

What is the exact length of the hypotenuse if the legs of a right triangle are 2 in. each? ​

A)
B)
C)
D)
E)

Question 50

Consider a square as shown in the figure. Find the total length of the four segments making up the stairs.
​ 7 in.
7 in.
​
__________ in.

Question 51

A man wanted to board a plane with a 5.4-ft-long steel rod, but airline regulations say that the maximum length of any object or parcel checked on board is 4 ft. Without bending or cutting the rod, or altering it in any way, how did the man check it through without violating the rule? ​

A) Put it horizontally.
B) Put it along the diagonal.
C) Put it vertically.
D) Put it along the width.
E) It is not possible without bending or cutting the rod.

Question 52

Approximate the irrational number with a rational number correct to the nearest thousandth. ​ ​

A) 9.870
B) 6.283
C) 3.000
D) 3.142
E) 1.571

Question 53

Simplify the expression. Assume that the variables are positive.
​

Question 54

Simplify the expression. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 55

Consider a square as shown in the figure. Find the total length of the four segments making up the stairs. ​
​ 5 in
5 in
​

A) 2.5 in.
B) 10 in.
C) 15 in.
D) 5 in.
E) 20 in.

Question 56

Approximate the irrational number with a rational number correct to the nearest thousandth.
​ __________

Question 57

An empty lot is 1,500 ft by 11,200 ft. How many feet would you save by walking diagonally across the lot instead of walking the length and width?
​
__________ ft

Question 58

Simplify the expression. ​ ​

A)
B) 0
C) ​
D) ​
E) already simplified

Question 59

An empty lot is 500 ft by 1,200 ft. How many feet would you save by walking diagonally across the lot instead of walking the length and width? ​

A) 0 ft
B) 500 ft
C) 400 ft
D) 1,700 ft
E) 609 ft

Question 60

Use the definition of square root to find the indicated product. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 61

Simplify the expression
​

Question 62

Completely reduce the fraction.
​

Question 63

Completely reduce the fraction.
​
​

Question 64

Perform the following operation.
​

Question 65

Perform the following operation. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 66

What is the exact length of the hypotenuse if the legs of a right triangle are 10 in. each?

Question 67

Perform the following operation. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 68

Completely reduce the fraction. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 69

Find an irrational number between 1 and 2.

Question 70

Simplify the expression
​

Question 71

Perform the following operation. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 72

Perform the following operation.
​

Question 73

Completely reduce the fraction. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 74

Perform the following operation. ​ ​

A) 20
B) -30
C) 5
D) -15
E) 15

Question 75

A unit fraction (a fraction with a numerator of 1) is sometimes called an Egyptian fraction. Egyptians expressed their fractions as a sum of distinct (different) unit fractions. How might the Egyptians have written the fraction ? ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 76

A man wanted to board a plane with a 4.3-ft-long steel rod, but airline regulations say that the maximum length of any object or parcel checked on board is 4 ft. Without bending or cutting the rod, or altering it in any way, how did the man check it through without violating the rule?

Question 77

Perform the following operation. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 78

Perform the following operation. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 79

Perform the following operation. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 80

Perform the following operation. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​