help me with this mindbender.?

Question

I once cashed a cheque at the bank. I had spent just one rupee from the amount before I realised the bank clerk had made a mistake. He had transposed the rupees with the paise. I now had exactly twice the value of the original cheque. What was the original cheques's value?
the answer is Rs.32.66
How did they come up with that, what's the trick in the problem and also what equations are to be used.
I Rs. = 100 paise.
thanks.

Answer 1

Let R be the number of rupees in the original check.
Let P be the number of paise in the original check.

The value of this check will be R*100 + P

Now if we transpose the digits, the new rupee value is P and the new paise value is R. In other words, the value of the check is P * 100 + R. We are told that 1 rupee (100 paise) was spent and the remaining amount is twice as big as the original.

As an equation:
(P*100 + R) - 100 = 2[ R * 100 + P]

100P + R - 100 = 200R + 2P

98P = 199R + 100

There is a hidden condition that you will also need:

0 <= R <= 99 and 0 <= P <= 99.

Now you need to find the smallest value of R that makes
199*R + 100 an exact multiple of 98.

R = 0 won't work, because the remainder on dividing by 98 is 2.
R = 1 won't work, because the remainder on dividing by 98 is 5.
R = 2 won't work, the remainder is 8.
R = 3 won't work, the remainder is 11.
R = 4, remainder 14
R = 5, remainder 17
R = 6, remainder 20
R = 7, remainder 23
etc.

Are you beginning to see a pattern? The remainder is 3R + 2. So when you get back to a multiple of 98 (namely 98) you'll have no remainder.

Now it is easy to solve:
3R + 2 = 98
3R = 96
R = 32

So you have 32 rupees. And from there you can calculate paise.

98P = 199R + 100
P = [199(32) + 100 ] / 98
P = (6368 + 100 ) / 98
P = 6468 / 98
P = 66

The final answer is that the original check was for Rs. 32.66 (and a double-check will confirm that Rs. 66.32 is 1 rupee more than twice the original)

Answer 2

okay, You have a cheque for Rs 32.66
you then encashed it for Rs 66.32
you then spent Rs 1
now you have Rs 65.32
divide that by two and you get . . . . .
. . .yes, Rs 32.66