## Tuesday, November 22, 2011

### this week's MGRE Math Beast Challenge

From MGRE:

This Week's Problem: "Modern Banking"

Q:

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:

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

Go to it. My answer will eventually appear in the comments, although another commenter's answer might appear there first. One remark before I go, though: that's got to be a pretty weird bank if the debit doesn't exactly match the deposit. I certainly wouldn't let any agency take two cents out of my bank account merely to verify that the account is mine!

_

Kevin Kim said...

The very first step is to make sure I've properly understood the problem. Let's pretend we're dealing with only ten customer accounts, and let's not worry about the 0.5% glitch situation quite yet.

If I'm an online bank like PayPal, and I'm doing this deposit/debit thing with ten customers, then I know that, for 7 customers, the d/d difference will be within the 2-cent range, and the remaining three differences could fall outside that range, to a maximum variation of 5 cents.

The variation can be either a profit or a loss for the bank. In terms of loss, then, the worst-case scenario would mean losing 2 cents for seven of the accounts, and losing 5 cents for the remaining three.

In real terms, for ten customer accounts, then, I could conceivably lose up to

[(7*.02)+(3*.05)] = [\$0.14 + \$0.15]

dollars, or \$0.29.

In fact, the above is true for every 10 customer accounts, isn't it? A maximum loss of 29 cents per 10 accounts? Let's keep that in mind.

If I'm dealing with 6000 accounts, but a glitch affects (i.e., nullifies) 0.5% of the attempted deposit/debit transactions, then how many of the accounts were successfully verified?

Well, 6000*.995 = 5970.

If I could conceivably lose a maximum of 29 cents per ten accounts, then I need to multiply \$0.29 times 597, the number of sets of ten accounts for which there were successful transactions (597 sets of 10 = 5970 accounts).

597*0.29 = \$173.13.

Hm.

That selection isn't offered, but (B) is looking like the best choice right now.

Where did I go wrong (or... did MGRE go wrong?)? Could it be that I've misread the problem? Have I misinterpreted the implications of the phrase "thus no money is transferred"? (I took it to mean there was neither profit nor loss.)

I'll come back to this problem later today. For now, I'll stick with (B) as the best possible choice.

Kevin Kim said...

5970*.3*.05 = \$89.55

5970*.7*.02 = \$83.58

\$89.55 + \$83.58 = \$173.13.

Even when I break the math down into smaller steps, I get the same answer.

Kevin Kim said...

.5% can be rewritten as .005.

.005*6000 = 30.

6000 - 30 = 5970.

Still getting 5970 accounts unaffected by the glitch. Is the problem my calculation of maximum potential loss? But I seem to be off by only a few cents. Let's go through this slowly, starting with the ten-accounts scenario:

Acct. 1: max. loss = .02
Acct. 2: max. loss = .02
Acct. 3: max. loss = .02
Acct. 4: max. loss = .02
Acct. 5: max. loss = .02
Acct. 6: max. loss = .02
Acct. 7: max. loss = .02

Acct. 8: max. loss = .05
Acct. 9: max. loss = .05
Acct. 10: max. loss = .05

Total maximum possible loss for all ten accounts = \$0.29.

I'm really not seeing where I've gone wrong.

Kevin Kim said...

I'm off from (B) by 27 cents. I'm off from (C), \$174.00, by 87 cents. Explanation...? Did I miss a trick somewhere, such that the answer is actually (A), (D), or (E)?

Still not seeing it.

Charles said...

This is probably not going to help, but I gave this problem a shot before looking at the comments and came up with exactly the same answer as you--\$173.13--by using roughly the same method.

\$174 would be the right answer if all the transactions went through. Technically, you should be able to then just take 99.5% of that amount to arrive at the correct answer: \$173.13. Like you, I used a more round-about way to originally arrive at my answer, but the logic here still holds. Right?

Or not. Whatever you're missing, I'm missing it, too.

Kevin Kim said...

I'm starting to think that MGRE goofed again, and that answer (B) is supposed to read "\$173.13."