Answer
Everyone got 5 points for this but not everyone paid attention to the guidelines. In particular, people spent much unnecessary time looking around for information on the web that they could find right on the exam page, or could figure out faster.
Some people didn't understand that they could take the exam home and work over the weekend to see if they could do better.
Answer.
I asked the Google calculator for
(2007-1776) years in secondsand it told me
(2007 - 1776) * years = 7.2896499 * 109 secondsThe United States is about 7.3 Gigaseconds old.
You can also solve this problem by doing the arithmetic yourself:
365 days 24 hours 60 minutes 60 seconds
231 years * -------- * -------- * ---------- * ----------
year day hour minute
= 7,284,816,000 seconds ~ 7.3 Gigaseconds
Google's answer and mine differ in the third decimal place because
they are more careful than I about the number of days in a year: it's
a shade under 365.25, not a simple 365.
-
Calculate your savings in dollars and as a
percentage of the original marked price.
Answer.
The new cost is
$64.99 * 0.50 * 0.85 = $27.62075 ~ $27.62
so I save $64.99-$27.62 = $37.38. The percentage savings is $37.38/$64.99 = 0.575 = 57.5%.You can also compute the percentage savings without computing the dollar savings first:
0.85 * 0.5 = 0.425
so you save1 - 0.425 = 0.575 = 57.5%.
- Filene's bought that item from their supplier for
$25.00. What is their profit as a percentage of their cost?
Answer.
The profit is $27.62 - $25.00 = $2.26. As a percentage of cost that's $2.26/$25.00 = 0.10483 ~ 10%.
-
What would their profit as a percentage of their cost be if
they'd sold the item at the original price?
Answer
At the original sales price the profit would have been $64.99 - $25.00 = $39.99. As a percentage of cost that's $39.99/$25.00 = 1.5996 ~ 160%.
This one's a little tricky. The profit is 160%, not just 60%!
- (Extra Credit) What is Filene's Basement's Automatic Markdown
System?
Answer From wikipedia (which seems pretty reliable for this kind of information)
A description of The Basement's markdown system from a 1982 New York Times article: "... every article is marked with a tag showing the price and the date the article was first put on sale. Twelve days later, if it has not been sold, it is reduced by 25 percent. Six selling days later, it is cut by 50 percent and after an additional six days, it is offered at 75 percent off the original price. After six more days - or a total of 30 - it is not sold, it is given to charity." Since the 1982 article was published the automatic markdown system was changed, giving more time between the discounts. There are now 14 days between each mark-down.
In fact a lot has changed since 1982 and Filene's Basement no longer offers an Automatic Markdown.
Student fees are about $3600/year for Massachusetts residents (umb.edu/administration_finance/bursar/tuition_fees.html#u_instate).
There are about 13,300 students (www.umb.edu/about/)
- What percentage of the operating budget is covered by student fees?
Answer.
In total, students pay ($3600/student)*(13300 students) = $47,880,000 in fees. That's $47.88 million/$237 million = 0.202025316 ~ 20% of the annual operating budget.
- If the annual inflation rate is 4.5%, what will next year's operating
budget need to be to offer the same services?
Answer. Next year's operating budget would have to be ($237 million)*1.045 = $247.67 million to offer the same services.
- The
easiest way for the Chancellor to raise all the extra money
is to raise student fees.
He knows that would be unpopular (and unfair), so wants to know how
much those fees would have to increase before he makes his decision.
If the total increase in expenditures is to be funded entirely by raising student fees, how much will each student's fees need to go up (give both an absolute answer in dollars and a percentage increase).
Answer.
The University will need $247.67 million - $237 million = $10.67 million more next year. Since there are 13,300 students, each will have to pay ($10.67 million)/13,300 = $802 more in fees. That would be an increase of $802/$3600 = 0.222848789 ~ 22%. Student fees would rise to to more than $4400.We could have predicted something like this large an increase without doing all the detailed arithmetic. Inflation will cause the budget to go up by 4.5%. Since student fees cover only about a fifth of the budget, they would have to increase by about 5*4.5% = 22% to make up the shortfall.
- The answer you've found in (c) depends on several approximations
and assumptions. It ignores the fact that some students have fee
waivers, some are
part time only, that
fees for out of state students and graduate students are larger, and
that the number of students may be different next year. How
reliable is your answer, given
these uncertainties? Could you trust it as a basis for deciding
whether to cover the budget shortfall just by raising fees,
or would you need a more accurate estimate?
Answer.
More students next year would make the added burden less. Taking out of state students into account would too. Computing the effect of fee waivers would make the average added burden more. So the answer isn't at all reliable in detail. But the fact that it tells the Chancellor that student fees would have increase by about 20% to make up the needed revenue tells him pretty clearly that he'd better look elsewhere for the money.
During the past year, the [Winslows] used just over 20,000 kilowatt-hours of electricity, of which 85% was consumed during off peak hours. Their power shifting saved them $348. ...Instead of paying the standard delivery charge of 6.2 cents per kilowatt hour all day long, the Winslows currently pay 13.95 cents from 9 a.m. to 6 p.m. weekdays and 3.4 cents from 6:01 p.m. to 8:59 a.m. on weekdays and all day weekends. Because they require a special meter, their fixed monthly customer charge is $9.99 instead of the standard $6.43.
- What would the Winslow's electric bill for the year have been
if they had not been participating in the experiment?
Answer
Here is how I computed the Winslow's bill:
20,000 kwh * 0.062 $/kwh + 12 months * 6.43 $/month = $1317.16.
But the 20,000 is only approximate so those pennies make no sense. Even the dollars are suspicious. The best way to report this is that it's about $1300 per year. - What was their electric bill?
Answer
The off peak electricity cost them
20,000 kwh * 0.85 * 0.034 $/kwh = $578.
(The 0.85 in that equation represents the fraction of the 20,000 kwh that they used during the off peak hours.) The peak electricity cost them20,000 kwh * 0.15 * 0.1395 $/kwh = $418.50.
Their total electric bill was$578 + $418.50 + 12*$9.99 = $1116.38or about $1100. - Do your calculations confirm what the article says about the
Winslows' savings?
Answer
No. I compute their savings as $1317.16 - $1116.38 = $200.78 ~ $200, which doesn't match the article's claim of $348 in savings. (It's appropriate to use the nonrounded values when taking the difference here since the two computations are probably approximate in the same way.)
It's hard to imagine why there's a mismatch. Perhaps the electric rates changed during the year and the article only reports what they were at some particular time. But it's clear that the Globe didn't do any arithmetic to check. Perhaps we'll write the author of the article a letter.
In its first 24 hours on sale, Harry Potter and the Deathly Hallows, the seventh and final installment in the wildly popular series by J.K. Rowling that officially went on sale at 12:01 a.m. Saturday, sold a record 8.3 million copies in the United States, according to Scholastic Inc., the book's publisher.
- On the average, how many books per minute were sold during
that first day?
Answer. There are 1440 minutes/day (Google told me) so the answer is about (8.3 million books)/day * (1 day)/(1440 minutes). Rather than do the arithmetic, I fooled Google into doing the unit conversion for me in one step. Although Google doesn't know about books/minute, it does know about miles/minute. So I searched for
8.3 million miles per day in miles per minuteand it told me8.3 million (miles per day) = 5763.88889 miles per minute.So Harry Potter sales were about 6000 books per minute or about 100 books per second for that first record breaking day. - Estimate how many bookstores were selling the book. (There's
no single correct numerical answer to this question. Be sure you
clearly state the assumptions you make that lead to your estimate.)
Answer.
I'll assume all the books were sold in retail stores and that sales continued at a steady pace all through the night and the next day. In fact those aren't reasonable assumptions. The 8.3 million copies probably includes millions that were ordered in advance on the web. And there were probably many hours when the bookstores were closed. But the question doesn't ask me to make good assumptions, just to state the ones I do make. With these assumptions the arithmetic isn't too hard.
I'll suppose it takes a minute for a sales clerk to ring up a sale. Then to process 6000 books per minute there would have to be 6000 cash registers in use all at the same time. If we estimate two cash registers per bookstore then there would have to be 3000 bookstores.
For a quick check I Googled "number of bookstores" and found out from www.laalmanac.com/arts/ar15.htm that there were 311 in Los Angeles in 2002. The population of Los Angeles was about 10 million in 2006 (quickfacts.census.gov/qfd/states/06/06037.html), which is 10/300 = 1/30 ~3% of the population of the United States. So assuming L.A. has just the right share of bookstores, there would be about 300/0.03 = 10,000 bookstores in the United States. That's not close to the 3,000 estimate I made, but it does have the right number of zeroes. It's close enough, given all the assumptions.
- (Extra Credit). In Harry Potter's wizard world currency is
measured in Galleons, Sickles and Knuts. 1 Galleon = 17 Sickles;
1 Sickle = 29 Knuts.
Scholastic Books, the publisher of the Harry Potter series in the United
States, has issued paperback copies of books that (they claim) Harry
Potter used at Hogwarts. The image shows the
price of one of these books in dollars and wizard money.
Use this information to figure out the number of Dollars per Galleon.
Answer.
First compute the price in knuts
29 knuts 14 sickles * -------- + 3 knuts = 409 knuts. sickleThere are17 sickles 29 knuts 493 knuts ---------- * -------- = ---------- Galleon sickle Galleonso$3.99 493 knuts $ --------- * ---------- = 4.81 ------- 409 knuts Galleon GalleonThe list price of Harry Potter and the Deathly Hallows is $34.99. How many Galleons is that? (You may write your answer with decimal fractions of a Galleon. No need to convert the fractional part to Sickles and Knuts - but you can if you want to.)
Answer.
1 Galleon 34.99 $ * --------- = 7.28 Galleons 4.81 $To change the fractional Galleons into sickles:17 sickles 0.28 Galleons * ---------- = 4.76 sickles. GalleonTo change the fractional sickles to knuts:27 knuts 0.76 sickles * -------- = 20.52 knuts sickleSo, finally, with some rounding$34.99 = 7 Galleons, 4 sickles and 21 knuts.
-
Burning a barrel of oil generates about 2000 kilowatt-hours of
useful energy
( instituteforenergyresearch.org/Energy_Conversion_Factors.htm). - The US imports about 10 million
barrels/day
(www.eia.doe.gov/neic/quickfacts/quickoil.html).
-
How much energy does burning our imported oil generate in a year?
Express your answer in metric terms with the proper prefix -
gigawatt-hours, or terawatt-hours, or petawatt-hours, or whatever is
appropriate.
Answer
kilowatt-hours barrels days 2000 ----------------- * 10 million -------- * 365 ---- barrel day year watt-hours = 2*103 * 103 * 107 * 365 ------------ year watt-hours = 730 * 1013 ---------- year Petawatt-hours = 7.3 -------------- year -
The oil we import isn't used primarily to generate electricity. But if
it were, how many families like the Winslows (see question 4) would it
provide electricity for?
Answer
1 family 7.3 Petawatt-hours * --------------------- 20,000 kilowatt-hours = (7.3/2) * 1015 * 10-4 * 10-3 families = 3.65 * 108 families ~ 370 million familiesSince there are only 303 million people, there aren't 370 million families. That proves that most of the imported oil is used for something besides domestic electricity! Probably transportation. -
Estimate how many oil tankers it would take to deliver the 10
million barrels of oil we import each day.
Answer.
The hardest part of this problem is figuring out how much oil a tanker can carry. According to www.generalmaritimecorp.com/newgencor5.html
ULCCs and VLCCs are the largest vessels in the world tanker fleet. They carry cargos of 200,000 dwt or greater. They typically transport oil in long-haul trades mainly from the Arabian Gulf to Western Europe and the United States via the Cape of Good Hope and Asia.
A dwt is a deadweight ton. Wikipedia says I might as well think of that as an ordinary ton: 2000 pounds or 1000 kilograms (those are close enough to one another for this kind of estimation). The web site www.eppo.go.th/ref/UNIT-OIL.html lists the weights of various kinds of oil. An average seems to be about 8 barrels per ton.So, finally
1 ton 1 tanker 107 barrels * ---------- * ------------ 8 barrels 200,000 tons = (1/(8*2)) * 107 * 10 -5 tankers = 0.0625 * 102 tankers ~ 6 (huge) tankerseach day can supply our imported oil.