Wednesday, June 23, 2010

WQ III:4 -- Story Problems

The Wednesday Quiz -- Season III -- Quiz 4

Story Problems

The Wednesday Quiz is a "closed-book" test of knowledge and intuition; please do not look up answers, ask others for help, or answer as a team.

Questions about the rules and the ~Fabulous Prizes~ are answered here.


It's time to revisit the horrors of the math classroom with this cunning series of eight story problems!

Here's the twist: for each problem, give both:

--> The Answer, and
--> Whether you used paper and pencil, used some sort of calculating device, or solved it in your head.

Problems correctly solved in your head are worth 13 points (to a maximum of 100); problems correctly solved on paper are worth 9 points; problems solved with a calculating device are worth 7 points. Lying on this point would of course make you an extremely bad person.

In recognition that this Quiz will potentially require more concentration than most, I'll leave it up until Friday evening.

Please note that this particular Quiz is awesome. Enjoy!
1. An octet and a quintet are travelling together by rail through a rainstorm. The train hits a stretch of railbed weakened by a rising stream, and derails. Half of the party, plus half a person, each lose a leg in the accident, although happily their arms are all fine and they do not lose their livelihoods. When they all meet on the anniversary of the disaster a year later, how many legs are present?

2. At 10 a.m., an airplane leaves Detroit for Dallas flying at 320 miles per hour. It reaches a cruising altitude of 36,000 feet shortly after take-off, but a half hour later the pilot notes a strange vibration in the right engine. Fortunately, he is able to solve the problem simply by adjusting the fuel flow. If there are 120 passengers on board, 60 percent of whom want cola, how many 12-can cases will the flight attendants need to have on hand?

3. Karl, a handsome, brooding young aristocrat, finds that he is irresistable to 25% of young women. If he himself is only attracted to 10% of young women, and there are only 200 women in all of Bavaria of a background and breeding sufficient to qualify as an eligible match, then statistics suggest he will most likely need to choose from among how many potential partners?

4. On the reading of old Mr. Pemberton's will, it is found that he has divided his estate, worth $30 million after taxes, rather capriciously. Angry Sarah, who ran away at fifteen, is to get one share of his estate; Thomas, the feckless diletante, 2 shares; perpetual student Maria, who married below her station, 3 shares; Brian, head of Marketing at PembertonCorp, 4 shares; and his favorite, Alice, the Executive Director of a leading orphanage, five shares. If Thomas burns through $200,000 a year, how long will he be able to live on his inheritance?

5. Jennifer needs to get out of town to let things cool down after a run-in with the cops, and is renting a car to drive 600 miles. The clerk at the rental agency suggests she "upgrade" from a car that gets 30 miles per gallon to one that gets 20 miles per gallon, for only $100 more. If gas costs $3.00 per gallon, how much more will Jennifer's trip cost if she accepts the upgrade?

6. A brave young hobbit encounters a enormous dog with three heads. He heroically attacks the hideous beast, but discovers to his horror that every time he slices off one of its heads, two heads immediately grow back to replace it! By the time he succumbs to the monster's savage jaws, it has ten heads. How many severed dog heads are strewn around the floor of the cave?

7. A young professor learns through the grapevine that he will only get tenure if he gets three articles published in major journals this year. As it happens, he has three articles ready for submission, but knows that for each article there is a 1/5 chance that it will be rejected in the peer review process, a 1/20 chance that it will be accepted by a journal that abruptly ceases publication before his article appears, and a 1/4 chance that it will be published but that a rival within the department will find an opportunity to sneeringly remark to the dean at a cocktail party that the journal in which it appears "could hardly be described as 'major.'" What are his odds of getting tenure?

8. You are one of two scofflaws in a conflict with Dirty Harry, who, in all the excitement, can't recall whether he has shot four or five of the six bullets that were originally in his gun. Assuming that he will choose randomly between the two of you to shoot at first if you elect to defy his will, knowing that Dirty Harry never misses, and considering that he is wielding a .44 Magnum, the most powerful handgun in the world, which would blow your head clean off, how lucky do you feel? Express your answer as a fraction, punk.

Submit your answers in the comments!


gS49 said...

1. 19-1/2 (head)--amputated at the knee?
2. 12 (head)--plus three litres of rum
3. 2-1/2 (head)--one's an amputee
4. 8 years (head)--or one year if he gets married
5. $130 (head)--but it's faster, so...
6. 7 (head)--he should have run away!
7. 1/400! (head)--time for a career change
8. 1/4 (head)--odds are better in prison (well, of surviving)

Elaine said...

1. The 'half-person' bit is just vague enough to cause unease, but the unscathed travelers still have all their legs (12) and I say there is one leg left on the 'half person' plus six legs left on the injured persons: 19 legs. Ick. I did that in my head but then got nervous and wrote 26 - 7 on the paper.

2. Oh, one of those 'irrelevant material included' questions. I used pen and paper to write 120 x .60= 72, which is how many inches a space-gobbling six-footer takes up. Six 12-packs would work, cola-wise, but I never drink the stuff.

3. He will be attracted to 20 women, but only 5 of them will reciprocate his interest, given your statistics. Unless he is just dumb enough to marry someone who really isn't very attracted to him, he's choosing from a group of 5. did it in my head, but did not brood over it.

4. Thomas's high-living ways will mean that his two shares will last him only ten years, the dumb-ass. I wrote down 1 2 3 4 5 = 15 and divided that into 30,000,000 to get 2 million. Move the decimal point over to divide by 10 and you see that equals Tommie's annual costs. I used pen on paper because it's too easy to get mixed up at a certain age.

5. Jennifer, having been stupid enough to commit a misdemeanor, which will be waiting for her when she comes back, would be lame-brained enough to think a gas-guzzler was an upgrade. She'll spend $100 for the upgrade and $30 more on gas, for a $130 increase in cost. IF, that is, she can write down the 600/20 and 600/30 problems and recall how to multiply by 3. Like I did.

6. Cerberus is certainly a menacing guard, eh? Seven lops later, the hobbit has a gory mess to clean up and a really serious need for stitches (or a coffin.) I drew a little lollipop diagram and kept score of the chopped heads below.

Elaine again said...

Egad, I missed two more questions!

7. Shoot, I don't know how to do this one. Probability was always painful, and I only got a little of it when teaching our son his 7th grade text over the summer...and he found 12 errors in the book. But I digress. Each article seems to have only about a 0.1% chance of publication, and he has to have all three published. Fageddabout it. I certainly intend to.

8. 1/4th. I took a wild shot at this and did it in my head.

Elizabeth said...

1. 26 legs leave on a voyage, less 6.5 legs = 19.5 legs, so apparently someone only lost half a leg. Half a leg is better than none.

2. 60% of 120 = 72, which is 6 cases.

3. 25% of 200 = 50. 10% of 50 = 5 potential partners.

4. $30 million divided into 15 shares = $2 million per share. Thomas has $4 million, so that will last him 20 years.

5. 600 miles @ 30mpg = 20 gallons x $3 = $60. 600 @ 20mpg = 30 gallons x $3 = $90 + $100 = $190 - $60 = $130 upgraded dollars.

6. 3 - 1 + 2 = 4
4 - 1 + 2 = 5
5 - 1 + 2 = 6
6 - 1 + 2 = 7
7 - 1 + 2 = 8
8 - 1 + 2 = 9
9 - 1 + 2 = 10
7 heads on the floor.

7. I'm not sure about the logistics of this, so I'm going to say that if there's a 1/5 chance it's rejected, there's a 4/5 chance it's not; a 19/20 chance it's accepted by a continuing journal; and a 3/4 chance that the dean is too drunk to remember what was said the next day, so 4/5 x 19/20 x 3/4 = 228/400 = 57/100 oh I don't know. I think there's a .57 chance that he'll get tenure.

8. I'm bored with fractions, so I'm going to pull this one out of the air and say that I'm only feeling 1/5 as lucky as I was before.

All work done as you see it, nothing up my sleeve, pay no attention to the mathematician behind the curtain.

The Calico Cat said...

While I can do this, I refuse! (I also will not take part in sentence diagramming.)

The Calico Cat said...

I officially vote for more cruise related posts. What did you eat - tomatoes? (any photos - aka food porn)For a start.

Anonymous said...

I see that I have fallen into at least one of M5000's evil traps. I liked it better when Thomas ran out of money in a year (which is why I put it on paper, because my brain no longer holds more than four digits at a time.

Cartophiliac said...

Does using my fingers count as a "calculating device"?

Cartophiliac said...

1. 13 legs (all in my head)
2. 6 (used paper)
3. 5 (best guess, using my head... assuming he is only limited to women of Bavaria. This is not explicitly stated, and we are not given enough information on the total number of women available in the rest of the world).
4. 10 years (used paper)
5. An upgrade would cost her $70 more (used paper)
6. 7 heads (used paper)
7. Snowball's chance in Hell (used my brain for that one)
8. 1/3 (on paper)

Low confidence in most of my answers. I didn't do well on story problems as a student either.

Elaine said...

No idea how I turned out as 'Anonymous' above. My Google log-in was absent (and it never works on this site) so I tried to put in my name. Stupid Mouse! (I'm sure it's not ME.)


Ben said...

I hope counting on my fingers counts as solving the problem in my head.

1. 19 legs (head)
2. Assuming the beverage cart only makes the rounds of the plane once, and that you're asking how many cases of COLA are required, and that none of the flight crew want any, then the flight attendants need 6 cases. (head)
3. He is not necessarily attracted to the same women as are attracted to him, so I'm going on the assumption that of the 10% he is attracted to, only 25% of THOSE return the favor. 5 women. (head)
4. 20 years. (head)
5. $130. (head)
6. 7 heads. (head)
7. I have the distinct feeling that I did not use the proper statistical formula to reach my conclusion of 57/100 (57%) chance of getting tenure. (head)
8. There's 1/4 chance that I'll be lucky, or to put it another way, 3/4 chance that I'll make Dirty Harry's day. (head)

Jenners said...

1. None ... everyone died on the way to the reunion when China nuked the country they lived in.

2. Only 2 ... they only accept cash and no one has any money.

3 0 Bavaria has no women.

4. 2 weeks ... then he is murdered by Angry Sarah

5. It will cost her a one way ticked to the slammer because the cops caught her based on the rental car clerk's description of her and the Lojack on the car.

6. None ... the dragon ate them.

7. 11% ... unless the dean figures out he slept with his wife ... then 0%.

8. 7/8ths lucky ... I am encased in bullet-proof Kevlar everywhere but on 1/8th of my body.

mrs.5000 said...

I did all of these in my head, because I'll be damned if I'm going to give up 4 points on each answer just for the convenience of a scratchpad. I was tempted to use Claymation in modeling #6, but held back.

1 19. Assuming they don't meet at the centipede convention, etc. etc.
2. 6 cases will give those poor cola-craving sods one can each. And, you know, I'm oddly relieved that the pilot solved that problem with the engine vibration.
3. 5, with the extremely dubious assumption that there is no correlation between W adoring K and K adoring W.
4. 20 years, unless he invests in tanzanite.
5. $130. But the car will "make it over the mountains."
6. I see seven. It's a grisly scene. Can't the hobbit adjust his fuel flow, or something?
7. Looks to me like 1 in 8. Is that why you left academia?
8. If he's shot 5, I've got a 1 in 2 chance that the scene will progress to fisticuffs or leaping across rooftops with me still alive. If he's shot 4, I'm dead. Since neither fisticuffs or roof-leaping are a strong suit with me, I'm feeling rather less than the computed one-quarter lucky.

This was fun, actually. Whether or not I do well, please praise me for "trying really hard."

Christine M. said...

Oh, criminy. I hate these things. I'm not going to attempt this one.

UnwiseOwl said...

1. I was put off for a second here wondering if like 'triplet', 'octet' and 'quintet' could be used to describe an individual person rather than an entire group, but then I realised that I was probably overthinking this.So there's thirteen people and 7 of them lose legs (plus half a person, bah). This leave us with 6 people with two legs (for a total of twelve), and 7 unipeds. Assuming that within a year, none of the unipeds have found their legs, gotten artificial limbs, and that none of them have partners that they've inexplicably brought (perhaps to lean on) that would give you 19 legs. I did this one in my head.

2. You are the master of the shaggy dog story, aren't you? If there are 120 people, and 72 of them want cola, you're going to need more cases than just the 6 needed to give all of those people one can. As well as having at least one case of spare cola for peopel to get seconds etc, since you also haven't specified what the cases are of they'll also need cases of lemonade, orange etc in the relevant proportions. I did this one in my head too.

3. Assuming that the variables that make our dashing aristocrat irresistable are evenly distributed with those that make him attracted to women, and assuming that there are no compromising factors such as hidden wives or suprise pregnancies...he has to choose between 5 women, perhpas it's time for a humiliating "Bachelor" series. Also in my head.

4. If Thomas is irresponsible and don't not invest the money at all,
then he has 1/15 of 30 million (2 million) which will last for 10 years. Head.

5. I had this one mixed up so that it ended up costing less when I did it in my head, so I resorted to paper and got $70 more...perhaps the salesperson is going to blackmail her into paying the extra?

6. Whoa dude, you just messed with a bunch of different mythologies there...10-3=7. In my head.

7. If he is unable to resubmit articles to another journal, and if his brown-nosing the dean gets him nowhere, and the dean listens to and respects the opinions of his faculty rivals; then our lecturer has a 50% chance of publishing each of his articles (I was almost fooled by the 1/5, but the number for success is 4/5, you cunning bugger). This means he has a 12.5% chance of publishing all three and getting tenure. Worth a try.I did this one on paper too. That means the odds are 1:7 if you need them expressed in a ratio.

8. There's a 1/4 chance that I'll survive the shooting, but given that if I got away he'd almost certainly be after me again, and Dirty Harry always gets his man...I'm not feeling very lucky at all...

UnwiseOwl said...

Oh, I did 7 on paper (using n/20) and 8 in my head. Fun quiz, Michael.

UnwiseOwl said...

Whoa, I didn't read the thing about scorign less for paper, that was dumb. And I screwed up...oh dear, not a good week for me...

chuckdaddy said...

1. 19
2. 6
3. 5
4. 20
5. 130
6. 7
7. 57/300 (not in head)
8. 3/4 chance I get shot

All in head (except for 7 which I think is wrong), but I did use my fingers a bit (does that downgrade the score even more than the other deductions?).

Elaine said...

I always skip the directions. I see the error of my ways, now. What the heck. AND my google identity is missing again below. This is tiresome.

Aviatrix said...

I just realized I typed a percent sign after my decimal answer to #7. Please count that as a typo and not as me not knowing how decimals work. Considering I'd just estimated it at 19 you have to see that, right?

Aviatrix said...

Damn, and number 8 says "fraction." Am I going to lose points for using a percentage?

Aviatrix said...

I wrote a long, complicated post showing all my work and commenting on the myriad unstated assumptions in the questions, but Blogger appears to have EATEN it, as my follow up comments have shown up but not my original. Here are just the answers, as I remember them.

No writing or other calculating aids except 7 where I used approximation in my head to get two decimal places and then my head hurt, so I used paper to get more, and 6 where I counted on my fingers for before I realized that might count as a calculation aid.

1. 19
2. 6
3. 5
4. 20 years
5. $130
6. 7
7. "0.19" in my head using estimation, "0.185166" on paper.
8. 1/4

Fortunately, the only answer I didn't remember was the one I used paper for and that one is written down.

fingerstothebone said...

1. How very gruesome -- 19 legs, in my head.
2. 6, in my head
3. Hmm, can not compute -- insufficient information in the question -- do the women have a chance to say no or is it purely based on his preferences?
4. 20 years, in my head.
5. Glad to see you put "upgrade" in quotes. $130, in my head.
6. 7, in my head (and fingers)
7. Hmm, can not compute -- insufficient information in the question -- what are the chances that the snide remark will make a difference to the dean and to the review panel's decision?
8. I'd tell him, "suppose a young professor learns though the grapevine that he will only get tenure if he gets three articles published in major journals this year...what are her odds of getting tenure?"

Jennifer said...

1. 6 [needed the words "octet" and "quintet" defined--"octet" indirectly through comparing it to "octagon"]
2. [this space intentionally left blank]
3 $30
4. 2 years (used paper)
5. $90.00
6. 7 heads (used paper)
7. [blank]
8. 1/4

[These answers prepared and submitted by JF for RF, who says, "Good Quiz! Tough challenge!"]

Michael5000 said...

1. An octet and a quintet means 13 people. Half of thirteen is 6 1/2, plus half a person -- in a (ripoff of/homage to) Lewis Carroll -- is seven legs missing. At the reunion, there will be 6 pairs of legs plus seven single legs, or NINETEEN legs.

2. Sixty percent of 120 passengers is 72; 72 cans of cola fits in SIX 12-packs.

3. For any given suitable woman, there is a (.25)(.10), or 2.5%, chance that Karl will be attracted to her AND that she will find him irresistable. 2.5% of the potential breeding pool yields a statistical likelihood of FIVE suitable mates.

4. The will divides the estate into (1+2+3+4+5), or 15, shares. Thomas thus ends up with 2/15 of the estate, or $4,000,000. At $200,000 per year, that's going to last him TWENTY years! Lucky boy, Thomas!

5. 600 miles at 30 mpg burns 20 gallons of gas; 600 miles at 20 mpg burns 30 gallons of gas. Ten extra gallons is $30, which when added to the $100 makes the trip ONE HUNDRED AND THIRTY DOLLARS more expensive.

6. Every head cut off yields a net gain of one head -- two new ones minus the one rolling on the floor. If the dog has gained seven heads at the end of the combat, it must have lost SEVEN heads.

7. The three factors that can result in our brainy friend's articles not counting towards his goal have a 4/20, a 1/20, and a 5/20 chance of occuring, leaving a 1/2 chance apiece that each paper will be published successfully. The odds that all three papers will be published are thus (1/2)(1/2)(1/2), or a 1/8 (12.5%) chance of getting tenure.

8. If you and the other scofflaw defy Mr. Harry's will, there is a 1/2 chance he will shoot at you first, blowing your head clean off. If not, there is a 1/2 chance that there will be a second bullet in the .44 Magnum, which will blow your head clean off. Ergo, you should feel (1/2)(1/2), or 1/4 lucky, punk.

Michael5000 said...

So, leading the charge with the only perfect score -- and with me enough of an ass to not only BE a little surprised but to confess it in print -- is Mrs.5000. Right behind her: Ben, Aviatrix, and a rare appearance from the elusive ChuckDaddy.

Also, we'll note that RF, with 22 points, becomes the youngest human ever to have scored points in the Wednesday Quiz!

gl. said...

i doubt i could have done any of these, but i very much want to see mrs.5000's claymation response to #6!

though i immediately knew the answer to #7 without doing the math: no chance at all, because publishing 3 articles in the next 6 months seems... unlikely, at best.

Aviatrix said...

Could you please explain why you get to add the three factors in number seven?

The way I see it is that each paper has to pass the peer review test, 4/5th chance, then the journal not failing test, a 19/20ths chance, and then the not being dismissed as a minor publication test, 3/4ths chance, so they have to be multiplied, not added.

And THEN that has to be cubed, to allow all three papers to run the gauntlet.

On any given day if there's a 25% chance of unsuitable weather, a 25% chance that I can't buy fuel, a 25% chance that a crewmember doesn't show up and a 25% chance that the customer cancels the flight, that doesn't mean that I never need show up to work, because there is an addtive 100% chance of not flying. Maybe 1/4 of the co-pilots solve it that way, leading to their no-show rate.

Michael5000 said...

Aviatrix: The difference in the two scenarios is that in your example the factors aren't mutually exclusive. Since any of them could happen in any combination, you've got a (.75)^4 chance of getting off the ground.

For the professor, the events are mutually exclusive. If his paper is rejected in the peer review process, it is by definition not accepted by a journal that may fail. If his paper is rejected in peer review or is accepted but never appears in print, his colleague has no reason to undercut him.

On a day that you have perfect weather, you still have the hurdle of fuel purchase ahead of you. But with an article that gets rejected in the peer review process, there are no more hurdles. It's this that makes the probabilities additive instead of multiplicative.

If you're like me and think about these things visually: your flight problem is a probability tree scenario; my professor's dilemna is a pie-chart-of-possible-outcomes sort of scenario.