103
Exam Math #2-Solutions
April 19, 2011
There are 11 problems and 2 extra credit problem.
Good luck and have fun! Show me all youve learned !
1. 10 points
Alex, Judy, and Lois are dividing the 18 inch long sub which has a 12 inch vegetarian
piece and a 6 inch meat piece. The sub is shown in the above picture. The three
friends are dividing the sub using the lone-chooser method.
Alex likes the vegetarian and meatball parts equally well.
Judy likes the meatball part four times as much as she likes the vegetarian part.
Lois likes the meatball part twice as much as she likes the vegetarian part.
Alex and Judy are the dividers and Lois is the chooser. In the first division, Judy divides
the sub into two shares (a left share s1 and a right share s2).
a. After Judy cuts, Alex gets to choose. Specify which of the two shares
Alex should choose, and give the values of the shares to Judy and Alex.
Value table
Vegetarian
Meatball
Alex
Judy
Lois
1 point
1 point
1 point
1 point
4 point
2 points
To Judy, the whole sub is worth 1*12+4*6*=36 points
Half is 18 points. All vegetarian is 12 points. In the meat part, we need
6 points.
6 points * 1 inch/ 4 points=1.5 inches
Judy cuts at the 12+1.5 inch mark from the left. 13.5 inch mark
Put your answers in the following table:
Shares
s1=[0 ,13.5]
s2=[13.5, 18]
value to Judy (as a fraction)
1/2
value to Alex (as a fraction)
13.5/18
4.5/18
a. Alex chooses s1 or s2?
Alex chooses s1 since it has a larger value to him.
b. Describe how Judy would then subdivide her share into three pieces. Use interval
notation.
There are 4.5 inches of meat in s2=[13.5,
18].
Dividing in 3 pieces, each piece is 1.5 inch.
The 3 pieces are : [13.5,15] [15,16.5] [16.5,18]
c. Describe how Alex would then subdivide his share into three pieces. Use interval
notation.
s1=[0 ,13.5]
Dividing in 3 pieces, each piece is 4.5 inch.
The 3 pieces are : [0,4.5] [4.5,9] [9,13.5]
True or false: Player 1 did not get her fair share. Hence she should get the 1 item left in the
surplus.
Explain your answer.
False.
Player 1 put her bids so that each segment is her fair share. So no matter which segment
she gets, she gets her fair share. The number of items in the fair share does not matter. It
is her fair share, according to her value system.
10 points
3. Three roommates, Sally, Linda, and Ann, are graduating from college and are dividing four
pieces of furniture using the method of sealed bids. Their bids on each of the items are given in
the following table. Complete this table, making sure to indicate in the last row of the table the
Items received and the Final cash settlement.
Alice
Desk
Refrigerator
Chair
Table
Total bids
Fair share
Value of items
received
Prelim Cash
settlement
Share of
surplus
Items received
and final cash
settlement
Bob
50
100
25
137
Carol
70
90
23
150
52
53
19
158
312
312/3=104
333
333/3=111
282
282/3=94
125
70
158
Pays 21
Gets 41
Pays 64
14.67
14.67
14.67
Refrigerator,chair
Pays 2114.67=$6.33
Desk
Gets 41+14.67=55.67
Table
Pays 64-14.67=$49.33
Surplus=21+64-41=85-41=44
44/3=14 2/3= 14.67
10 points
Over several days, the price of a toy did the following: on Monday the
price went down by 17%, on Tuesday it went up by 7.2%, on Wednesday it
went down by 3.4%.
5.
a. With the decreases and increase in price, after the 3 days, was there a
percentage increase or a percentage decrease in the price of the toy?
(1-.17)(1+.072)(1-.034)= 0.8595
.8595 is less than 1. So there is a percentage decrease.
b. What is the percentage increase or decrease? Round your answer to 1
decimal place.
1-.8595=.1405=14.1%
c. If the original price of the toy is $47, what is the final price of the toy?
Round your answer to 2 decimal places.
0.8595*$47=$40.40
5 points
6. Suppose that you deposit $345 in a savings account with an APR of
3.25%, compounded daily (interest is credited the to account at the end of
each day). Find the balance in the account after 4.5 years. Round your
answer to 2 decimal places.
F=P(1+r/n)^nt
P=345
r=.0325
n=365 t=4.5
nt=4.5*365=1642.5 which rounds down to 1642
The bank gives interest at the end of the day. so parts of a day dont
count toward interest.
F=345(1+.0325/365)^1642=$399.31
5 points
7. a. What is APY an abbreviation for?
Annual percentage yield
b. Goodrates Bank advertises an APR of 4.60%, compounded monthly.
Find the APY at this bank. Round your answer to 2 decimal places.
F=1(1+.0460/12)^12=1.04698233
APY=4.70%
c. An APR of 4.60% compounded monthly is equivalent to an APR of
4.70%, compounded annually.
8. 15 points
Starting at the age of 20, you put $100 at the beginning of each month into a bank with an APR
of 5.75% compounded monthly. You will continue doing this until you are 65 years old. (You will
withdraw the money when you turn 65.) Assume that the bank pays interest at the end of each
month.
Round your answer to 2 decimal places.
a. What is the future value of your first $100. deposit when you turn 65?
Part a is a compound interest problem
Use F=P(1+r/n)nt
with P=100, r=.0575, n=12, and t=65-20=45
F=P(1+r/n)nt=100*(1+.0575/12)12*45=100*(1+.0575/12)540=132
1.457
Answer is $1321.46
b. YOU DONT NEED TO DO THE CALCULATION IN PART b. You just need to set up the
expression that you would calculate.
What is the future value of your second $100. deposit when you turn 65?
Part b is a compound interest problem
Use F=P(1+r/n)nt-1
with P=100, r=.0575, n=12, nt-1=539, as the
money is in the bank for month less than it is for the first $100.
deposit.
That is, we use 12*45-1=540-1=539
F=P(1+r/n)nt=100*(1+.0575/12)539
c. YOU DONT NEED TO DO THE CALCULATION IN PART c. You just need to set up the
expression that you would calculate.
What is the future value of your last $100. deposit?
Part c is a compound interest problem
Use F=P(1+r/n)1
with P=100, r=.0575, n=12, and the exponent is
1, as the money is in the bank for only 1 month.
F=P(1+r/n)1=100*(1+.0575/12)1
d. How much money will be in the account when you turn 65?
Round your answer to 2 decimal places.
Use the geometric sum formula.
P=future value of last deposit=100*(1+.0575/12)
c=1+.0575/12 N=12*45=540
So the sum equals:
P * (cN-1)
=
c-1
P * (cN-1)
c-1
= 100*(1+.0575/12)*((1+.0575/12)540-1)
1+.0575/12-1
=
256134.40
e. YOU DONT NEED TO DO THE CALCULATION IN PART e. You just need to set up the
expression that you would calculate.
If, instead of putting the money in the bank at the beginning of each month, you put it in at the
end of each month, how much money would be in the account when you turn 65?
Use the geometric sum formula. P=future value of last deposit=100
c=1+.0575/12
N=12*45=540
So the sum equals:
P * (cN-1)
c-1
= 100*((1+.0575/12)540-1)
1+.0575/12-1
=
254912.9
1 point
Extra credit problem 1.
A bank has an APY of 4.50%
It has an APR of r, compounded daily.
Set up, but do not solve, an equation in which you can find the value of r.
Hint: Think how, starting with an APR, you find an APY.
Solution:
Find the future value of $1 for 1 year. Subtract $1. Convert to a percentage This is the
APY.
Given the APY=.045, add 1. This gives 1.045
Find the future value of $1 for 1 year at an APR of r, compounded daily.
1(1+r/365)^365=1.045
2 points. 1 for proper setup and 1 for the correct answer.
Extra credit problem 2.
Sam is saving for a used car that he wants to buy in 4 years, after graduation from college.
He needs $7,000. He will put $x in the bank each month, on the first of the month, for the next 4
years. He found a bank that has an APR=5.25%, compounded monthly.
How much money does he need to put in the bank each month so that he will have $7,000 in 4
years?
Solution:
P=x(1+.0525/12)
C=(1+.0525/12)
N=48
7000=x(1+.0525/12)*((1+.0525/12)^48-1)/(.0525/12)
Answer is $130.80