Susan cashed a check at the bank but the teller made a mistake. he gave her as many dollars as he should cents, and as many cents he should've given her dollars. Susan took the money without realizing the error and spent $3.50. She then had twice as much money as the actual check. What was the amount of the check?
One way, using a similar problem statement:
Sarah goes to the bank to cash a check for D dollars and C cents. The bank teller mistakenly gives her C dollars and D cents (so C and D must both be whole numbers between 0 and 99). Sarah doesn't discover the mistake until after she has spent 59 cents. She then notices that the remaining money is twice the actual value of the check. What was the value of the check?
1--Let the check be for D dollars and C cents.
2--Then we can write the check amount as D + .01C or 100D + C.
3--The amount received then becomes 100C + D.
4--Then, 100C + D - 59 = 2(100D + C).
5--This simplifies to 98C - 199D = 59.
6--Dividing through by the lowest coefficient, we get C + 2D + 3D/98 = 59/98.
7--(3D + 59)/98 = C - 2D
8--(3D + 59)/98 must be an integer.
9--Multiplying by 33 and dividing through by 98 again we get D + D/98 + 19 + 85/98
10--(D + 85)/98 must also be an integer.
11--Set (D + 85)/98 = k or D = 98k - 85.
12--Substituting back into (5) we get 98C - 19502k + 16915 = 59 or C = 199k - 172.
13--Since C and D can be no more than 2 digit numbers, only k = 1 produces a valid result.
14--For k = 1, C = ? and D = ?.
15--Therefore, the original check was for $?.
16--The amount received was $?.
17--Check: $? - .59 = 2($?).
