What is formula for solving a transposition problem please read discussion for the actual problem?

Here's the problem as you posed it: : A woman goes to the bank to cash a check. The bank teller makes a mistake. She transposes the dollars and cents, and gives the woman way too much. On the way home, the woman loses $0.02 through a hole in the bottom of her purse. She now has exactly twice the face value of the original check. (We're assuming here that she had no money when she went to the bank.) : I already know the answer to this question. The answer is $32.65 (and $65.32), but that is not what I want here. What I want to know is how would you solve this problem without just trying numbers all day until you accidentally hit on the answer? Let D = correct number of dollars and C = correct number of cents So, if we work in cents: * 100D + C was the correct amount of money. * 100C + D was the amount given in error.

We can write: (100C + D) - 2 = 2(100D + C) which will simplify... 100C + D - 2 = 200D + 2C to this: 98C = 199D + 2 We have here two variables and only one equation, so we cannot find the solution directly. However, for the problem to make sense D and C must be two-digit integers (whole numbers) and this condition will enable us to solve it. We need to consider exactmultiples of 98 and of 199. * N.B. 2 x 98 is 196. 196 is three less than 199. Therefore every time we increase C by two, the next multiple of 199 will climb 3 further away. Just to illustrate this: * 2 x 98 = 196 = 1 x 199 - 3 * 4 x 98 = 392 = 2 x 199 - 6 * 6 x 98 = 588 = 3 x 199 - 9* 8 x 98 = 784 = 4 x 199 - 12

We want our multiple of 98 to be 2 higher than the nearest multiple of 199; so the previous multiple of 98 would have been 96 lower: C x 98 = D x 199 + 2 Subtract 98 from this: (C - 1) x 98 = D x 199 - 96Look back to the examples above, taking in the pattern, and you will appreciate that this equation will be satisfied when D = 96/3* = 32, and C - 1 = 64. Therefore D = 32 and C = 65; the amount was $32.65. That method was not of course a formula; it was more of an algorithm to speed up the process of guessing a bit. Thanks for the interesting question! *We were fortunate here that 96/3 turned out to be a whole number, bringing us swiftly to a solution.