document.write( "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? \n" ); document.write( "

Algebra.Com's Answer #601432 by KMST(5328) $\"\"$ $\"About$

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$\"x\"$ = number of whole dollars in the check amount.
\n" ); document.write( " $\"y\"$ = number of cents in the check amount.
\n" ); document.write( " $\"100x%2By\"$ = value of the check amount, in cents.
\n" ); document.write( " $\"100y%2Bx\"$ = amount given by the bank teller, in cents.
\n" ); document.write( " $\"100y%2Bx-5\"$ = amount taken home (after spending a nickel at a gum machine), in cents.
\n" ); document.write( "\"When I returned home, I found I had exactly twice the amount of the check\" translates as
\n" ); document.write( " $\"100y%2Bx-5=2%2A%28100x%2By%29\"$
\n" ); document.write( "We work with that equation to find a solution that makes sense for the story in the problem.
\n" ); document.write( "We need a pair of non-negative integers (x,y), with $\"y%3C=99\"$ .
\n" ); document.write( "Solving:
\n" ); document.write( " $\"100y%2Bx-5=2%2A%28100x%2By%29\"$
\n" ); document.write( " $\"100y%2Bx-5=200x%2B2y\"$
\n" ); document.write( " $\"100y%2Bx-5-2y-x=200x%2B2y-2y-x\"$
\n" ); document.write( " $\"98y-5=199x\"$
\n" ); document.write( " $\"98y-5%2B5=199x%2B5\"$
\n" ); document.write( " $\"98y=199x%2B5\"$
\n" ); document.write( " $\"y=%28199x%2B5%29%2F98\"$
\n" ); document.write( " $\"y=%28196x%2B3x%2B5%29%2F98\"$
\n" ); document.write( " $\"y=196x%2F98%2B%283x%2B5%29%2F98\"$
\n" ); document.write( " $\"y=2x%2B%283x%2B5%29%2F98\"$ .
\n" ); document.write( "The equation above is a linear equation with an infinite number of (x,y) pairs,
\n" ); document.write( "but to fit the story the solution pair (x,y) must meet stringent requirements.
\n" ); document.write( "We need a pair of non-negative integers (x,y), with $\"y%3C=99\"$ .
\n" ); document.write( "For $\"y%3C=99\"$ to be a non-negative integer,
\n" ); document.write( " $\"%283x%2B5%29%2F98\"$ must be a non-negative integer with $\"%283x%2B5%29%2F98%3C=99\"$ ,
\n" ); document.write( "meaning that $\"3x%2B5=98n\"$ for some non-negative integer $\"n\"$ ,
\n" ); document.write( "and $\"98n%3C=99\"$ <--> $\"n%3C99%2F98\"$ .
\n" ); document.write( "So, we are limited to $\"n=0\"$ and $\"n=1\"$ .
\n" ); document.write( " $\"3x%2B5=98%2A0=0\"$ gives us a negative fraction for $\"x\"$ , so it does not work.
\n" ); document.write( " $\"3x%2B5=98%2A1=98\"$ ---> $\"3x=98-5\"$ ---> $\"3x=93\"$ ---> $\"x=93%2F3\"$ ---> $\"highlight%28x=31%29\"$ .
\n" ); document.write( "Then, $\"system%28y=2x%2B%283x%2B5%29%2F98%2C3x%2B5=98%2Cx=31%29\"$ ---> $\"y=2%2A31%2B1\"$ ---> $\"y=62%2B1\"$ ---> $\"highlight%28y=63%29\"$ .
\n" ); document.write( "The exact amount of the check was $\"highlight%28%22%2431.63%22%29\"$ . \n" ); document.write( "

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