Question

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? (Note: 1 dollar = 100 cents.)

Answer 1

The check was for $18.56.

Let x be the number of dollars in the check, and y be the number of cents. Consider the numbers of dollars and cents Ms Smith holds at various times. The original check is for x dollars and y cents. The bank teller gave her y dollars and x cents. After buying the newspaper she has y dollars and x − 50 cents. We are also told that after buying the newspaper she has three times the amount of the original check; that is, 3x dollars and 3y cents.

Clearly (y dollars plus x − 50 cents) equals (3x dollars plus 3y cents). Then, bearing in mind that x and y must both be less than 100 (for the teller's error to make sense), we equate dollars and cents.

As −50 less than or equal to (x − 50) less than or equal to 49 and 0 less than or equal to 3y less than or equal to 297, there is a relatively small number of ways in which we can equate dollars and cents. (If there were many different ways, this whole approach would not be viable.) Clearly, 3y − (x − 50) must be divisible by 100. Further, by the above inequalities, −49 less than or equal to 3y − (x − 50) less than or equal to 347, giving us four multiples of 100 to check.

If 3y − (x − 50) = 0, then we must have 3x = y, giving x = −25/4, y = −75/4
If 3y − (x − 50) = 100, then (to balance) we must have 3x − y = −1, giving x = 47/8, y = 149/8
If 3y − (x − 50) = 200, then we must have 3x − y = −2, giving x = 18, y = 56
If 3y − (x − 50) = 300, then we must have 3x − y = −3, giving x = 241/8, y = 747/8

There is only one integer solution; so the check was for $18.56.

Answer 2

yes!