## Tuesday, February 09, 2010

### Check, Please

A confused bank teller transposed the dollars and cents when he cashed a check for Ms Smith, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Smith noticed that she had left exactly three times as much as the original check. What was the amount of the check?

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brain teaser

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$18.56

ReplyDeletewe ARE extremely rusty in our

ReplyDeletealgebraic skills. :-)

we spent an inordinate (pun!)

amount of time juggling

x's, y', a' & b's with nothing

to show for it except a proper humbling.

{our guess is that it was overdue}

we were able, however,

to confirm Andy's answer through other means.

it appears that $18.56 is correct. kudos!

and thanks for the twister

..

.ero

Let x be the number of dollars in the check, and y be the number of cents.

ReplyDeleteThen 100y + x − 50 = 3(100x + y).

Therefore 97y − 299x = 50.

Keep in mind that x and y have to be between 0 and 100

It can be shown that all integer solutions of 97y − 299x = 50 are of the form y = 1850 + 299k, x = 600 + 97k, where k is any integer. (I'm not going to show this... too much writing!)

In this case, because x and y must be between 0 and 99, we choose k = −6.

This gives y = 56, x = 18.

So the check was for $18.56.

In case anyone cares to know, the way I found this one was sort of a brute force method. I got the equation that Mike mentioned, and solved for x:

ReplyDeletex = (97y - 50)/299

Then I used a quick spreadsheet to find x for all possible values of y (00...99), and found that x=18, y=56 was the only one that came out to integer values, so that was my answer.

Not sure if the spreadsheet is considered cheating though :)