Switched dollars and cents

Question

A bank teller switched the dollars and cents when he cashed a check for a customer, giving him dollars instead of cents and cents instead of dollars.

After putting 4 quarters into a vending machine, the customer discovered that he now had exactly twice as much as his original check.

What was the amount of the check?

Answer 1

Let's say the amount of the check was x dollars and y cents, whose total value would be = 100x+y cents

After switching dollars and cents, the customer got y dollars and x cents, whose value = 100y+x cents

Now, he spent 4 quarters (= 100 cents) in a vending machine

So, he got left with (100y+x-100) cents, which, according to the question, is twice the amount of real check amount

i.e. 100y+x-100 = 2*(100x+y)

98y = 199x+100

y = (199x+100)/98

y = (196x+3x+98+2)/98
y = 2x+1+(3x+2)/98        eq1

[Below it would become clear why I am writing eq1 in this form]

Now, x and y can only be positive whole numbers <100

It is clear from the equation that y will be positive whole number when 3x+2 will be divisible by 98

i.e. 3x+2 = 98 will be one of the solutions

In that case, x = 32 and y = 2*32+1+1=66, so the check amount was $32.66

 

The next solution for eq1 would be when, 3x+2 = 2*98

In that case, x=66

and, y = 2*66+1+2=135, which does not make sense because y being cents should be <100

Similarly, for all solutions of eq1 such that 3x+2 = n*98, where n>=2 would result in either x or y or both being >100

Therefore the only solution is that the paycheck amount was $32.66