MATHEMATICS 100 Name:
MATHEMATICS 103 Name: KEY
Section 01
EXAM 03
November 7, 2008
This exam has 3 pages, including this cover page. Please make sure you have all 3 pages.
You have 55 minutes to complete this exam.
You are allowed to work with one partner. If you do work with a partner, just turn in one exam.
This exam is open book and open notebook. You may use your scientific calculator, and you may log-in to the computer at your desk and use a blank Excel spreadsheet or the Windows calculator, but do not print anything out. You are not allowed to use any other application, such as email, a web-browser, instant-messaging, or personal electronic devices such as a cell-phones or MP3 players. There is a "Zero-Tolerance" policy for violation of this policy. That is if you use any of these applications during this exam, it will be considered a violation of the code on Academic Integrity and you will receive 0 points.
You must show all your work for full credit. Check your answers wherever possible.
The answers you handwrite below will be graded. Do not print anything out.
For the problems below, make sure I know which formula(s) you used and what all the quantities are.
1. A student saves $3,400 from his summer job and wants to invest it. He invests it in an account at an annual interest rate of 2.12% compounded monthly, and doesn’t add or withdraw any money during the year.
[3 pts] How much will be in the account at the end of one year?
Compound Interest A = P * (1 + APR/n)^n*t
A = 3400 * (1+0.0212 / 12)^12*1
A = 3400 * (1.0017667)^12
A = 3400 * 1.0214
A = $ 3,472.78
[3 pts] How much interest will the student earn for the entire year?
Total interest = Total at the end of the year – total at the beginning = 3,472.78 – 3,400 = $ 72.78
[3 pts] What is the annual percentage yield (APY) of the account?
Find interest after one year and divide by the initial principal.
Interest in first year = 72.78 (see answer above)
APY = 72.78 / 3,400 = 0.0214 = 2.14% note that this answer was embedded in the work above
Now, we would like to consider the interest on a monthly basis.
[2 pts] First, calculate the amount of interest the student earned in the first month.
Use the Compound Interest formula, but note that t=1/12, for one twelfth of a year, A = P * (1 + APR/n)^n*t
A = 3400 * (1 + 0.0212 / 12)^12*1/12 = 3400 * (1 + 0.0212 / 12)^1 = 3400 * (1.0017667)
= 3,406.006667 rounded to 3,406.01 ( $ 6.01
[2 pts] Next, calculate the amount of interest the student earned in the second month.
Previous result * (1 + 0.0212 / 12) = 3,406.01 * 1.0017667 = 3,412.02 ( 3,412.02 – 3406.01 = $ 6.02
[3 pts] Finally, how would you explain the trend from your previous two answers? If the amount of interest earned each month is getting larger, why?
$6.01 the first month, $6.02 the second month. Since the savings formula is exponential, it would be concave-up, and so the growth would be at an ever increasing rate.
2. Suppose you want to accumulate $10,000 for a down payment on a home, in four years, by making regular end-of-the month savings deposits. Assume that you find an account with an APR of 3%, compounded monthly.
[3 pts] What formula would you use to help determine how much should you deposit at the end of each month to reach your goal? Only give the formula, and then plug in what you know. You do not need to calculate anything.
By the way, if you were to complete the calculation, you would find that the monthly deposit should be $196.34.
Accumulated Savings A = PMT * ( (1+APR/n )^n*t – 1) / (APR/n)
10,000 = PMT * ( (1+0.03 / 12 )^12*4 – 1) / (0.03 / 12)
For completeness…
10,000 = PMT * ( (1.0025)^48 – 1) / (0.0025) = PMT * (1.127328-1) / 0.0025 = PMT * (0.127328) / 0.0025
10,000 = PMT * 50.9312 ( P = $ 196.34
[3 pts] How much of this accumulated amount comes from the deposits, and how much comes from interest (assuming the 3% rate stays constant)?
196.34*12*4 = $ 9,424.48 deposited.
That leaves 10,000 – 9,424,48 = $ 575.52 earned from interest
3. [4 pts] After graduating, you decide to purchase a “Hummer” that’s been retro-fitted to be eco-friendly: bio-diesel, hybrid technology, etc. Since it’s an experimental vehicle, the sale price is relatively reasonable at $33,250. If you find a loan with an APR of 1.9% interest, how much would you need to pay each month to pay off the balance over a period of time of five years? Installment Loan Formula, PMT = p*(APR/n) / ( 1 – (1+APR/n )^-n*t )
PMT = 33,250*(0.019/12) / ( 1 – (1+0.019 / 12)^(-12*5) )
PMT = 33,250*0.00158333 / ( 1 – (1.001583333)^-120 ) = 33,250*0.00158333 / (0.090559)
PMT = $581.34
4. The country of QMerica has three states, which are The QS, Apathy, and Estado. The population of each state is given below. The leadership committee of QMerica consists of 9 seats.
[2 pts] Calculate the standard divisor. Total pop / # seats = 30 / 9 = 3.3333
[8 pts] If the 9 seats are apportioned using Lowndes’ method, how many representatives would each state have? Fill in the table below.
|State |Population |Standard Quota |Relative |# of Seats Using |
| | | |Fractional Part |Lowndes’ Method |
| | | 16 / 3.333 | 0.8 / 4 | |
|The QS |16 |= 4.8 |= 0.2 |4 |
| | |5 / 3.3333 |0.5 / 1 | |
|Apathy |5 |= 1.5 |= 0.5 |1 + 1 = 2 |
| | |9 / 3.3333 |0.7 / 2 | |
|Estado |9 |= 2.7 |= 0.35 |2 + 1 = 3 |
Alternatively, the number of seats could be apportioned using the Huntington-Hill method. Pick one appropriate modified divisor, and use it to fill in the table below. Note that you do not need to apportion all 9 seats by repeatedly trying new divisors, just do one iteration with the divisor you choose.
[1 pt] Which modified divisor do you choose? Larger than standard divisor above, 3.5
[9 pts] Fill in the table for one iteration of the Huntington-Hill method. Answers will vary, depending on divisor
| | | | | |# of Seats Using the |
|State |Population |Standard |Modified |Geometric |Huntington-Hill |
| | |Quota |Quota |Mean |Method |
| | | | | |after ONE iteration |
| | | 16 / 3.333 | 16 / 3.5 |sqrt(4*5) | mod quota > geo mean, |
|The QS |16 |= 4.8 |= 4.57 |= 4.47 |round-up |
| | | | | |5 |
| | |5 / 3.3333 |5 / 3.5 |sqrt(1*2) |mod quota > geo mean, |
|Apathy |5 |= 1.5 |= 1.429 |= 1.414 |round-up |
| | | | | |2 |
| | |9 / 3.3333 |9 / 3.5 |sqrt(2*3) |mod quota > geo mean, |
|Estado |9 |= 2.7 |= 2.57 |= 2.4495 |round-up |
| | | | | |3 |
5. The students on a floor of Quanster Hall reported the number of hours they spend on social networking web-sites, per day: 1.0, 4.0, 2.0, 0.5, 1.4, 2.1, 2.8, 0.75.
[2 pts] Give the median of the data. [1 pt] What is Q1 of the data? [1 pt] What is Q3 of the data?
Sort: 0.5, 0.75, 1.0, 1.4, 2.0, 2.1, 2.8, 4.0
Median: avg. of the 4th and 5th positions Q1 is the median of the smallest 4 Q3 is the median of the largest 4
(1.4 + 2.0) / 2 = 1.7 ( 0.75 + 1.0 )/2 = 0.875 ( 2.1 + 2.8 )/2 = 2.45
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| | | |
| | | |
|Page Number |Possible Points |Score |
| | | |
| | | |
|2 |26 | |
| | | |
|3 |24 | |
| | | |
| | | |
|Total |50 | |
| | | |
Honor Pledge
I am familiar with the policy for Academic Integrity as outlined in the Pathfinder:
(intranet.juniata.edu/policies/pathfinder/acadhonesty.html).
I have not discussed the contents of this exam before it has been administered to me, and I will not discuss the contents of this exam before it is administered to someone else.
I understand that failure to comply with this agreement will constitute cheating and subject me to a charge of academic dishonesty.
_______________________________
Signature
_______________________________
Date
Project 02 is due on Monday, 10.Nov
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