In reality, the bill was about 68.37 dollars. Mr. Brown should have received 31.63 in change. Due to the mix up of cents and dollars, he received an amount of 63.31. After losing 5 cents, he still had 63.26; which is double the correct amount of change mentioned above.
You will get the correct solution by setting up a linear equation:
- X = correct dollar amount
- Y = correct cent amount
X + Y : 100 he should have received.
But, he has got back Y + X : 100.
If you subtract 5 cents from this amount the double sum of the correct amount (mentioned above) is the result:
X : 100 + Y - 0,05 = 2 * (X + Y : 100)
After transposing the equation you have:
Y = (199 : 98) * X + 5 : 98
This expression represents a linear equation. The associated definition area is 0 < X < 100 (as he paid with a 100 dollar bill).
X and Y are natural numbers, thus, only integer values are possible.
This means, in the interval 0 < X < 100, the result is X = 31 and Y = 63. Mr. Brown should have received 31 dollars and 63 cents back from the waitress. The check was about 100 - 31.63 = 68.37 dollars.