Tuesday, November 29, 2011

MGRE's solution to last week's Math Beast Challenge

Remember last week's MGRE Math Beast Challenge problem?

The answer is indeed B, so I would have gotten it right, but according to the answer write-up, the amount of $173.40 is correct-- it's not $173.13. Personally, I'm not so sure, even after having read MGRE's explanation. But you decide: here's what they had to say.


An online bank verifies customers’ ownership of external bank accounts by making both a small deposit and a small debit from each customer’s external account, and asking the customer to verify the amounts. In 70% of these exchanges, the deposit and debit are within two cents of one another (for example, a deposit of $0.18 and a debit of $0.16, or a deposit of $0.37 and a debit of $0.38), and the deposit and debit are always within five cents of one another. During one week, the online bank attempts to verify 6,000 accounts in this manner, but 0.5% of the transactions do not go through, and thus no money is transferred. What is the maximum amount, in dollars, that the account verification system could have cost the bank that week?

(A) $165.30
(B) $173.40
(C) $174
(D) $256.71
(E) $258


This is just a very lengthy problem that requires careful reading and note taking.

Of 6,000 accounts, 70% have deposits and debits 2 cents apart, and the other 30% have deposits “within 5 cents” (but not within 2 cents), and thus are 3-5 cents apart. So:

4,200 are 1-2 cents apart
1,800 are 3-5 cents apart

0.5% (that’s one-half of one percent) of 6,000 attempts do not go through, so:

30 do not go through
5,970 do go through

We are not told how many of the 30 failed attempts were in the 1-2 cents apart category and how many were in the 3-5 cents apart category. However, we are trying to MAXIMIZE the bank’s cost, so we’re going to finish this problem by presuming the worst possible scenarios for the bank. To maximize the loss, presume that:

• All the (2-5 cent) differences are in the customer’s favor
• All the costs are as large as possible (so the 1-2 cent ones are all 2 cents, and the 3-5 cent ones are all 5 cents)
• The 30 accounts that did not go through were the two-cent ones (that way we can maximize the 5-cent losses)

Thus, we WOULD have had:

4,200 2 cent losses
1,800 5 cent losses

...except for the 30 exchanges that didn’t go through. Again, to maximize the bank’s loss, let’s assume that the 30 that didn’t go through were 2-cent losses. Therefore:

4,170 2 cent losses = $83.40
1,800 5 cent losses = $90

$83.40 + $90 = $173.40

The correct answer is B.

While the explanation sounds plausible, I don't agree with the idea that the failed transactions should be counted as losses. I don't see this implied anywhere in the problem, which to my mind makes MGRE's assumption unwarranted. As a practical matter, though, that's just a quibble because my own reckoning puts me in the proper ballpark. I agree with the MGRE gurus that (B) is the best answer of the bunch.



Charles said...

"While the explanation sounds plausible, I don't agree with the idea that the failed transactions should be counted as losses."

I'm not sure what you mean by this. The failed transactions are not, in fact, counted as losses. Maybe I'm misunderstanding this sentence?

Where we differed from MGRE was in our assumptions about the distribution of the failed transactions.

There were 30 total failed transactions. If the distribution of these transactions were even, then 21 (70%) would be 2-cent losses and 9 (30%) would be 5-cent losses:

(4200 - 21) * 0.02 = 83.58
(1800 - 9) * 0.05 = 89.55

Add these two figures together and you get our original result, 173.13. What MGRE is saying, though, is that if we are looking for the most money the bank could have lost, we should assume that the failed transactions are all of the less expensive variety, as opposed to being equally distributed.

This makes sense to me. I just failed to take that little fact into account.

Kevin Kim said...

Well, the MGRE explanation says "let’s assume that the 30 that didn’t go through were 2-cent losses." I assumed, in my calculations, that the failed transactions were neither profits nor losses: transactions had been attempted in those 30 cases, but no transactions had actually occurred. How, then, assume that those transactions represent a loss? They're neither a loss nor a gain.

Anyway, once I had dismissed those 30 transactions from my calculations, the rest followed.

Charles said...

Hmm. I see where you're coming from. I guess the confusion boils down the the phrase "go through," which is frankly rather vague. What exactly does this mean?

The way I saw it (after reading the explanation), these 30 failed transactions would have been 2-cent transactions had they gone through. What exactly that means, I don't know.

All in all, I think this is a rather retarded problem that derives its difficulty from its obfuscatory nature.