Question 980224: I went to the bank to cash a check. By mistake, the teller gave me dollars instead of cents and cents instead of dollars, which I did not notice. On the way out, I spent a nickel at a gum machine. When I returned home, I found I had exactly twice the amount of the check. Can you calculate the exact amount of the check I cashed?
Found 2 solutions by ankor@dixie-net.com, KMST:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! I went to the bank to cash a check.
By mistake, the teller gave me dollars instead of cents and cents instead of dollars, which I did not notice.
On the way out, I spent a nickel at a gum machine.
When I returned home, I found I had exactly twice the amount of the check.
Can you calculate the exact amount of the check I cashed?
:
let a = correct dollar amt
let = correct cents in decimal form
;
100b + .01a - .05 = 2(a + b)
100b + .01a - .05 = 2a + 2b
100b - 2b = 2a - .01a + .05
98b = 1.99a + .05
b = 
Using the table feature on my Ti83, found only one solution (y =  )
a = 31, b = .63
Correct amt of check $31.63
You got $63.31
you spent .05
---------------
leaving $63.26 Which is twice as much as 31.63
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website!  = number of whole dollars in the check amount.
 = number of cents in the check amount.
 = value of the check amount, in cents.
 = amount given by the bank teller, in cents.
 = amount taken home (after spending a nickel at a gum machine), in cents.
"When I returned home, I found I had exactly twice the amount of the check" translates as
We work with that equation to find a solution that makes sense for the story in the problem.
We need a pair of non-negative integers (x,y), with  .
Solving:
 .
The equation above is a linear equation with an infinite number of (x,y) pairs,
but to fit the story the solution pair (x,y) must meet stringent requirements.
We need a pair of non-negative integers (x,y), with .
For to be a non-negative integer,
 must be a non-negative integer with  ,
meaning that  for some non-negative integer  ,
and  <--> .
So, we are limited to  and  .
 gives us a negative fraction for , so it does not work.
 ---> ---> ---> ---> .
Then,  ---> ---> ---> .
The exact amount of the check was  .
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