Honors MAT111

Test II - Quantitative

Fall 2002

  1. Suppose that at the end of you junior year in High School, one of your grandparents gave you $10,000 to help pay for a new car when you graduate from college.  You aren’t allowed to use the money until you graduate.
    1. Suppose you can earn 6% compounded annually in a certificate of deposit, if you lock in the rate for 5 years.  To what amount would it accrue?

A = 10,000(1 + .06)5                        = $13382.26

    1. Now, suppose you could choose instead to earn 5.875% compounded monthly, if you lock in the rate for 5 years.  To what amount would the $10,000 accrue in this case?

A = $10,000(1 + .05875/12)60        = $13,404.87

    1. Your third option is daily compounding at 5.79%.  What is the best choice?

A = $10,000(1 + .0579/365)5 x 365 = $13,357.29

 

The best choice is b.

  1. Suppose you decide to begin saving for retirement early.  You get some help from a rich Aunt who began contributing $200 a month to a plan when you turned 15.  When you graduate from college exactly 6 years later, she will stop contributing and hope that you continue to do so.
    1. If her plan averaged 6.5%, how much would have accrued by the time you graduate from college?

A = $200[(1 + .065/12)72 – 1] / (.065 / 12) = $17,554.23

    1. Now suppose that you after you graduate from college, you continued the $200 monthly contributions until you retire on your 70th birthday. How much will be in the account under the following three scenarios

                                                               i.      6% annual rate?

A1 = $200[(1 + .06/12)12 x 49 – 1] / (.06 / 12) = $711,111.31

A2 = $17,554.23(1 + .06/12) 12 x 49 = $329,629.52

 

Total = $1,040,740.83

                                                             ii.      7% annual rate?

A1 = $200[(1 + .07/12)12 x 49 – 1] / (.07 / 12) = $1,013,844.75

A2 = $17,554.23(1 + .07/12) 12 x 49 = $536,641.09

 

Total = $1,550,485.84

                                                            iii.      8% annual rate?

A1 = $200[(1 + .08/12)12 x 49 – 1] / (.08/ 12) = $1,462,471.13

A2 = $17,554.23(1 + .08-/12) 12 x 49 = $873,306.05

 

Total = $2,335,777.18

  1. Your finances get tight in the year before you graduate from college and you rack up $5000 in credit card debt. 
    1. If the rate is 9.99%, determine what your monthly payments would be if you want to pay off the balance in

                                                               i.      Five years

Pmt = $5,000(.0999/12) / (1 – (1 + .0999/12)-60) = $106.21

                                                             ii.      Three years

Pmt = $5,000(.0999/12) / (1 – (1 + .0999/12)-60) = $161.31

                                                            iii.      One year

Pmt = $5,000(.0999/12) / (1 – (1 + .0999/12)-60) = $439.56

    1. What would you have paid to retire the principle in each case?

                                                              i.      $6,372.60

                                                            ii.      $5,807.16

                                                            iii.      $5,274.67

  1. You have finished college and are working at your first job earning a take-home pay of $2200 each month.  You can get a mortgage in which the payments do not exceed 33% of your monthly take-home pay.
    1. What is the maximum home price you can afford at the following rates on a 30-year loan? 

 

.33 x 2200 = $726

                                                               i.      7%

P = $726[1 – (1 + .07/12)-360)] / (.07/12)     = $109,123.29

                                                             ii.      6.5%

P = $726[1 – (1 + .065/12)-360)] / (.065/12) = $114,861.05

                                                            iii.      6%

P = $726[1 – (1 + .06/12)-360)] / (.06/12)      = $121,090.71

    1. What would the home have cost you in each case?

$261,360

    1. Suppose after you got the loan, you get a promotion and your take home pay increases to $3000 a month.  If you decide to turn your 30-year mortgage into a 15 year mortgage by increasing the payments, what would your monthly payments be in each of the cases in part a?

 

$3,000 x .33 = $990

 

                                                               i.      P = $990[1 – (1 + .07/12)-360)]   / (.07/12)    = $148,804.49

                                                             ii.      P = $990[1 – (1 + .065/12)-360)] / (.065/12) = $156,628.71

                                                          iii.      P = $990[1 – (1 + .06/12)-360)]   / (.06/12)    = $165,123.70

What amount of your payment on a $100,000 mortgage at 6.5% goes towards principal in the following cases? 

Pmt = $100,000(.065/12) / (1 – (1 + .065/12)-360) = $632.07

Pmt = $100,000(.065/12) / (1 – (1 + .065/12)-180) = $871.11

                                                           iv.      First payment, 30-year loan?

Int     $632.07 / (1+.065/12) = $628.66

Prin. $632.07 - $628.66 = $3.40

                                                             v.      First payment, 15-year loan?

Int     $871.11 / (1+.065/12) = $866.42

Prin. $871.11 - $866.42= $4.69

                                                           vi.      120th payment, 30-year loan?

Int     $632.07 / (1+.065/12)120 = $330.55

Prin. $632.07 - $330.55 = $301.52

                                                          vii.      120th payment, 15-year loan?

Int     $871.11 / (1+.065/12)120 = $455.56

Prin. $871.11 - $455.56 = $415.55

  1. Give an example that illustrate each of the following situations
    1. An event with probability 0

Rolling two dies and getting a sum of 0 or 13, 14, 15, etc.

    1. An event with probability 1

Rolling two dies and getting a sum between 2 and 12 inclusive

    1. An event with probability ¼

Tossing a fair coin twice and getting exactly two heads, or getting exactly two tails 

  1. You toss a fair coin five times
    1. Draw a tree diagram

  

    1. Circle the results on the tree diagram in the following cases

                                                               i.      Exactly two heads.  What is the probability of getting exactly 2 heads?

  

                                                             ii.      Exactly three heads.  What is the probability of getting exactly three heads?

  

  1. You toss a pair of fair dice
    1. What is the probability of getting a sum of seven?

  

 Look at the chart to find 6/36 possibilities.

  

    1. What is the probability of getting a sum of five?

Sum of 2 or 12 = 1/36

Sum of 3 or 11 = 2/36

Sum of 4 or 10 = 3/36

Sum of 5 or 9   = 4/36

Sum of 5 or 8   = 5/36

Sum of 7           = 6/36

  

  1. You get a box of chocolates as a Christmas present.  15 of the pieces are creams, which do can’t stand and there are only 5 that you like – with nuts in them.  Unfortunately, you can’t figure out a way to tell them apart.
    1. What is probability on selecting a piece with nuts?

P(chocolate w/ nuts) = 5/20 = ¼

  

    1. What is the probability of selecting at least one piece containing nuts if you select

                                                               i.      Two pieces?

Use the “at least one rule”

 1 – (15/20)2 = .4375                          

                                                             ii.      Three pieces?

  

   1 – (15/20)3 = .578125                       

  1.  
    1. If the probability of it snowing on a given day in January is .1, what is the probability that is will snow at least once in the month of January?

                       

                        1 – (.9)31 = .9618 

    1. What is the probability of drawing a Queen or a heart from a standard deck of 52 cards?

P(Q or ©) = 4/52 + 13/52 – 1/52 = 16/52 = 4/13 

    1. How confident would you be that you could produce a Queen or Heart 30 times on 100 draws from a deck of 52 cards?

From the above solution, 4/13 = 30.77%.  So, you’d expect to get a Queen or Heart 30 times out of 100, but looking at figure 7.6 on pg. 429 you shouldn’t be that confident about it. 

  1.  You find yourself in Atlantic City one evening with $200 dollars in your pocket
    1. If you place 40 five-dollar bets on black at the roulette table, what is the amount with which you’d expect to walk away?

The expected value of each bet is E.V.(Black) = (-$5)(18/38)  + ($5)(16/38) = 26¢

                        (26¢ ) x (40 Bets) = $10.40 

    1. Suppose you’ve learned to play blackjack well enough that you have maximized your changes of winning to 49.3%.  If you play 40 five-dollar hands, what would is the amount with which you’d expect to walk away?

The expected value of each bet is E.V.(BlackJack) = (-$5)(.507)  + ($5)(.493) = 7¢

                        (7¢ ) x (40 Bets) = $2.80 

  1.  Suppose the national death rate for all types of cancers is .2%.  If you are studying the data of towns with a size 300,000 and notice two in which 1,500 people annually are dying of cancer, would you be suspicious of environmental factors?

You would expect only about 600 cancer deaths.  Or you could note that 1500/300,000 = .5% 

  

  1. Derive the savings plan formula.  Bonus +5 for deriving the loan payment formula.