SGU – Episode #410 – Who’s That Noisy?
Question
A bank teller made a mistake today. The teller switched the dollars and cents when they cashed a check for Mrs. Jones, giving her dollars instead of cents and cents instead of dollars.
After buying a newspaper for 5 cents, Mrs. Jones realized that she had remaining exactly twice as much as the original check.
What was the amount of the original check?
Answer
Let original check be x dollars and y cents. So the money given is y dollars and x cents. Therefore:
100y + x – 5 = 2(100x + y)
So,
199x – 98y = -5
We require integer values of x and y that solve this equation (a Diophantine equation). Following the algorithm to solve a Diophantine equation:
199=2*98+3
98=32*3+2
3=1*2+1
2=2*1
1=3-1*2=3-1(98-32*3)=33*3-1*98=33(199-2*98)-1*98=33*199-66*98-1*98=33*199-67*98
(* denotes multiplication).
So, a=33 and b=67 solves the equation 199a-98b = 1. Multiplying both sides by -5, we get:
199(-5a)-98(-5b) = -5.
Hence, x = -5×33 = -165 and y = -5×67 = -335, solves the equation 199x – 98y = -5. However, these are not the only solutions. The general solution is given by:
x = -165+98n and y = -335+199n, where n is an integer.
Check:
199(-165+98n)-98(-335+199n) = -199*165+199x98n+98×335-98x199n = -199×165+98×335 = -5
We require a solution where both x and y are positive. The first solution of this kind occurs when n=2, giving:
x = -165+98*2 = 31 and y = -335+199*2 = 63.
Hence, the original check was 31$63c.
arielerodino 
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