**Name**

This is a closed book, closed note exam. It consists of 8 problems, plus

3 extra credit problems. Each problem is worth 12 points, and each extra

credit problem is worth 3 points. In addition, you will receive 4 points

for writing your name. You may not use a calculator on this exam. Good luck!

1. Alexandra's used car lot is offering an additional 15% o cars that are

already 20% o the original price. Meanwhile, Mira's used car lot is

advertising 35% o the original price for all the cars on the lot. If the

car you want is available at both places (at the same original price),

where should you go and why? How much will you save by going to

the one used car lot as opposed to the other?

2. (a) Which of the numbers below are natural numbers? Which are

integers? Rational numbers? Real numbers? Fill in the chart

below by writing "yes" if the number belongs to the corresponding

set, and "no" if not. (Note: a number may belong to more than

one of these sets.)

(b) What is the additive inverse of -5? Why?

(c) There is one rational number that does not have a multiplicative

inverse. What is it? Why doesn't this number have a multiplicative inverse?

(d) Write a word problem that corresponds to this arithmetic problem:

(-6) (-2/3 ).

3. (a) Write the number 81 in base 6.

(b) Write the number 103_{5} in base 10 (recall that the subscript 5

indicates that the number is currently written in base 5).

(c) Do the following base 5 arithmetic problem: 30001242_{5}×21_{5}. (You

may leave your answer in base 5.)

4. (a) Peter is just beginning to learn to count using our
base 10 enumerative system.

He doesn't understand why 30 is the next number

after 29. How would you explain it to him?

(b) Which of the four fundamental operations of arithmetic
satisfy

the commutative property?

(c) What is 200,000 + 600,000?

(d) What is 21+43?

(e) Recall the distributive property of multiplication: if a, b, and c are numbers, then a(b + c) = ab + ac. In your solutions to problems (c) and (d) above, you implicitly used this property. Explain how.

5. (a) Determine the prime factorization of 231.

(b) What is the smallest number with 7 factors? What is
the smallest

number with 6 factors?

(c) Haley's comet can be seen every 75 years. The
Swift-Tuttle comet

can be seen every 135 years. If one year we see them both at the

same time, how long will it be before they are both visible in the

same year again?

6. For each of the four problems below, select the model
from the list that

ts. After brie y justifying your choice, solve the problem.

Models: | + Combine, increase - Take-away, comparison, missing addend ×Repeated addition, area, Cartesian product Partitioning, repeated subtraction, missing factor |

(a) Julia is preparing a meal. She knows how to cook 13 different

entrees and 17 different side dishes. If a meal is a combination

consisting of one entree and one side dish, how many different

possible meals can Julia cook?

(b) In Portland, Maine, it was -11 degrees Farenheight on December

21st. After 8 days, the temperature warmed up to -5 degrees

Farenheight. How much warmer was it in Portland on December

29th than it was on December 21st?

(c) My car has an average fuel efficiency of 43 miles per gallon. If I

can travel 417.1 miles on a tank of gas, how large (in gallons) is

my gas tank?

(d) One day, Elizabeth decides to walk home from the office. If the

total distance from her office to her home is
miles, and she has

walked 2/3 of the way, how far away from the office is she?

7. (a) On Josh's birthday, his mom cuts his birthday cake into 6 pieces

and gives him 1. On Sammy's birthday, his mom cuts his cake

into 12 pieces and gives him 2. Josh begins to cry because Sammy

got more pieces of cake on his birthday than Josh got on his. How

would you explain to Josh that he actually got the same amount

of cake as Sammy?

(b) On Jerry's birthday, his mom cuts his birthday cake into 20 pieces

and gives him 3. Now Sammy is crying because Jerry got more

pieces of birthday cake than he did. How would you explain to

Sammy that he actually got more cake than Jerry did?

(c) Now that Sammy knows that he got more cake than Jerry, he

wants to know how much more. Determine how much more cake

Sammy got than Jerry, and then describe how you would explain

your solution to Sammy.

8. Let a; b be two numbers such that 0 < b < a < 1. In the table below,

fill in the second column with either <, = or > so that the first three

columns give a true number sentence. In the fourth column, give a

justification for your choice of <, =, or >. (Note: examples do not

constitute sufficient justification and will not receive full credit.)

**Extra Credit:** What is your instructor's favorite number?

**Extra Credit:** What is the last digit of 9^{100}?

**Extra Credit:** Write the following rational number
in the form a/b ,

where a and b are integers: 0.027027027027027 …