Chapter 1 & 2 Test [100 points] Due Week 2 Day 7 (Sunday)
Name___________________________________
Please save the file as LastName_Chpt_1&2_Test.doc.
For each question, type your work/explanation and solution in the appropriate place. Use Equation Editor to type any
mathematical expressions or equations. Each question will be scored using the following rubric.
5
4
3
2
0
Solution is correct.
Solution is correct.
Solution is not correct, Solution is correct, but No solution or work
Solution is fully
Solution is supported
but explanation and/or no explanation and/or given.
supported by accurate by explanation and/or mathematical
mathematical
Solution is incorrect
explanation and/or
mathematical
calculation is shown.
calculation is given.
and no work is shown.
mathematical
calculation. Minor
calculation.
mistakes in
calculation may have
been made.
*1-2 points may be deducted for not using equation editor
SHORT ANSWER.
Write the word or phrase that best completes each statement or answers the question.
1) How are the symbols
3
, 0.75, and 75% related?
4
SOLUTION
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The symbols shown above all represent the same quantity.
3
, 0.75, and 75% are just different
4
ways (fraction, decimal and percentage, respectively) of expressing the real number “three fourths”.
They are all equal to each other and can therefore be used interchangeably.
2) Give a verbal, visual, numerical, and graphic representation for the idea one and one-sixth.
SOLUTION
a)
b)
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Verbal: one 6-pack and one extra can (of soda)
Visual:
c) Numerical:
d) Graphical:
7 1
, 1 , 1.16666666..., 1.16
6 6
3) Provide the missing representations.
SOLUTION
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Visual:
Numerical:
7 1
, 2 , 2.333333..., 2.3, 233.33%
3 3
Graphic:
Provide an appropriate response.
4) Explain why no one-to-one correspondence can be established between the elements of {4,7,9} and {w,x,y,z}.
SOLUTION
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There are 3 elements in {4,7,9} and four elements in {w,x,y,z}. A one-to-one correspondence
between two finite sets it is possible if and only if they have the same cardinal numbers. (same
number of elements). The reason is that for every element of {4,7,9} should correspond a unique
element in {w,x,y,z} and viceversa. This is clearly impossible.
5) Use the concept of addition to show that 8 > 2.
SOLUTION
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Start from the valid inequality 6>0. Adding a 2 to both sides we obtain 6+2>0+2 that is 8 > 2.
6) A is the set of all the letters of the alphabet and B is the set of vowels. What kind of relationship exists between the two
sets? Also, if C is the set of consonants what is the relationship between B and C?
SOLUTION
The set of vowels, B, is contained in the set of letters, A. In set notation: B  A. On the other
hand, the set of consonants the set of vowels are disjoint from each other. In set notation:
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B  C  .
7) U is the universal set and B is a proper subset of U. Write a relationship between the cardinal numbers of U, B and B .
SOLUTION
Denote by n(X)- the cardinal number of a set X. Then, we have:
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n( B)  n( B )  n(U )
8) Illustrate with an example that (a  b)  c  a  (b  c) in the set of whole numbers.
SOLUTION
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(3+5)+6 = 3+(5+6) is equivalent to 8+6 = 3+11 that is, 14=14, true. This is an illustration of the fact
that addition is associative.
9) Given that n(P) = 10 and P ⊂ Q , what is the least number of elements that set Q can have? Is there a maximum limit
on the number of elements that set Q can have?
SOLUTION
Q is going to have at least 10 elements. Actually, if  is used to denote strict inclusion, then Q
has to have at least 11 elements. There is no maximum limit on the number of elements of set Q.
10) In the set of whole numbers, is
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4  9 meaningful? Explain.
SOLUTION
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If by the set of whole numbers we understand the set of all nonnegative integers than the operation
4-9 does not make sense. The result would be -5 which does not belong to the set of whole
numbers. This shows that the set of nonnegative integers is not closed under subtraction.
MULTIPLE CHOICE. Choose the letter that best completes the statement or answers the question. Show all
work and explain your solution completely.
11) Use inductive reasoning to predict the next number in the sequence. 1, 
A) 
1
1024
B)
1
4096
C) 
1
4096
D)
1 1
1
1
, , ,
, ... .
4 16
64 256
1
1024
WORK/REASONING
We notice that each term is obtained by multiplying the previous one by 
SOLUTION
1
.
4
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A)
1
. The next term is therefore
4
1
1
1
going to be obtained by multiplying
by  . We thus obtain 
.
256
4
1024
These terms form a geometric sequence with ratio 
12) Solve the problem and explain the problem-solving strategy used. How many different amounts of money can you pay
if you use four coins including quarters, nickels, and dimes?
A) 12 amounts B) 15 amounts C)14 amounts D) 13 amounts
WORK/REASONING
SOLUTION
List all possible cases:
(4,0,0) 4 quarters, 0 dimes, 0 nickels --- 100 cents
(3,1,0) 3 quarters, 1 dime, 0 nickels ----- 85 cents
(3,0,1) 3 quarters, 0 dimes, 1 nickel ------ 80 cents
(2,2,0) 2 quarters, 2 dimes, 0 nickel ------ 70 cents
(2,1,1) 2 quarters, 1 dime, 1 nickel ------ 65 cents
(2,0,2) 2 quarters, 0 dimes, 2 nickels ------ 60 cents
(1,3,0) 1 quarter , 3 dimes, 0 nickel ------ 55 cents
(1,2,1) 1 quarter, 2 dimes, 1 nickel ------ 50 cents
(1,1,2) 1 quarter, 1 dime, 2 nickels------ 45 cents
(1,0,3) 1 quarter, 0 dimes, 3 nickels------ 40 cents
(0,4,0) 0 quarters, 4 dimes, 0 nickels----- 40 cents
(0,3,1) 0 quarters, 3 dimes, 1 nickel----- 35 cents
(0,2,2) 0 quarters, 2 dimes, 2 nickels----- 30 cents
(0,1,3) 0 quarters, 1 dime, 3 nickels----- 25 cents
(0,0,4) 0 quarters, 0 dimes, 4 nickels----- 20 cents
C) 14 amounts
There are 15
combinations in
total but two of
them amount to
the same total.
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13) Indicate whether the sequence is arithmetic, geometric, or neither. 10, 12, 22, 32, 42, . . .
A) Geometric B) Neither
C) Arithmetic
WORK/REASONING
SOLUTION
Look first at the successive differences: 21-10=2, 22-12=10, 32-22=10, 42-32=10.
The sequence is not arithmetic since not all the successive differences are the
same. Then look at the successive ratios: 21/10 =6/5, 22/12=11/6 …. The sequence
is not a geometric sequence since not all the successive ratios are the same.
B)
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14) Solve the problem and explain the problem-solving strategy used. A dad is now three times as old as his son, but in
eleven years the dad will be only two times as old as his son is then. How old will the dad be next year?
A) 44 B) 33 C) 43 D) 34
WORK/REASONING
SOLUTION
Denote dad’s current age by D and his son’s present age by S. The problem tells us
that D=3S and that D+11 =2(S+11). Substituting D in the second equation we get
that: 3S+11=2S+22 from which S=11, D=33. Eleven years from now the ages are
going to be 44 and respectively 22. Dad’s 33 now so next year he’ll be 34.
B)
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15) Solve the problem and explain the problem-solving strategy used. Joe has a collection of nickels and dimes that is
worth $4.35. If the number of dimes were doubled and the number of nickels were increased by 10, the value of the coins
would be $7.05. How many dimes does he have?
A) 22 dimes
B) 11 dimes
C) 10 dimes
D) 43 dimes
WORK/REASONING
SOLUTION
Denote by d = the number of dimes and by n= the number of nickels in Joe’s
collection. We know that 10d+5n=435 (we converted everything to cents)
On the other hand 20d+5(n+10)=705. We need to solve this two equations, two
unknowns system. First simplify the first equation by dividing by 5: 2d+n=87.
Same thing for the second equation: 4d+n=131. Subtracting the first equation from
the second term by term we get: 2d =44 that is, d=22.
A)
16) Convert the number to the indicated base.
A)16five
B) 55five
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24six to base five
C) 31five
D) 131five
WORK/REASONING
SOLUTION
24six = 2x6+4 =16ten=31five. The argument for the second step is the following:
16 divided by 5 gives a quotient of 3 and a remainder of 1.
A)
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17) Identify n(S) for the given set. S = {0}
A) 2
B) ∅
C) 0
D) 1
WORK/REASONING
SOLUTION
In this case S consists of one element (element 0). So n(S)=1
D)
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18) Write the specified statement. State the contrapositive of the following:
If he is happy, then he is smiling.
A) If he is smiling, then he is happy.
B) If he is not happy, then he is not smiling.
C) If he is happy, then he is not smiling.
D) If he is not smiling, then he is not happy.
WORK/REASONING
SOLUTION
If he is not smiling, then he is not happy.
D)
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19) Determine whether the sets are equal only, equivalent only, both equal and equivalent, or neither equal nor
equivalent. {first, second, third} and {1, 2, 3}
A) Equal
B) Equivalent C) Both
D) Neither
WORK/REASONING
SOLUTION
The sets are not equal since they consist of different elements; for instance “first”
and “1” may suggest the same thing but they are nevertheless different entities.
B)
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However, the sets are equivalent since they both consist of exactly three elements.
20) Write the statement indicated. Write the negation of the following:
The test is difficult.
A) The test is not difficult.
B) The test is not very easy.
C) The test is very difficult.
D) The test is not easy.
WORK/REASONING
SOLUTION
The test is not difficult.
A)
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