Alan

Note:  1 Watt is not 1 Joule second; it is 1 Joule per second.

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I just fitted a 4th order polynomial to the sequence.  \(f(n)=\Sigma_{k=0}^4 a_kn^k\)

There are five unknowns (the a_k) and five numbers in the sequence.

Peter's marbles initially = p

Harry's marbles initially = h

p/h = 4/5

p+11 = h - 8

From the first equation p = 4h/5.  Put this into the second

4h/5 + 11 = h - 8          Subtract 4h/5 from both sides and add 8 to both sides

19 = h/5               Multiply both sides by 5

h = 95               Put this into p = 4h/5 to get  

p = 4*95/5   or  p = 76

Peter had 76 marbles at first.

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One possibility is 245:

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143.3333 = 1.02^x   Take logs of both sides and use the property of logs that  ln(a^b) = b*ln(a)   (where ln is log to the base e)

ln(143.3333) = x*ln(1.02)    Divide both sides by ln(1.02)

x = ln(143.3333)/ln(1.02)  

x = 250.7330

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500 = -500/3*t + 200       Divide all terms by 100

5 = -5/3*t + 2                  subtract 5 from both sides

0 = -5/3*t -3                    Multiply all terms by 3

0 = -5t - 9                      Add 5t to both sides

5t = -9                          Divide both sides by 5

t = -9/5

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buttons (b) = red (1/4b)     +   yellow(1/4b +32)    +   blue(86)

         b = (1/4)b  +  (1/4)b+ 32  + 86

         b = (1/2)b + 118

        (1/2)b  =  118

              b = 236

See if this helps:

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I'm puzzled (I often am!).  What do you mean by "normal" math here?  In what way doesn't this involve algebra?

Not clear why you don't want algebra!  You could try a couple of guesses and iterate towards an answer, or, perhaps plot a graph like the following, where 3times (Jack's original amount less 150) and Kenny's original amount less 45 is plotted against Jack's original amount.  Where the two lines cross tells you Jack's original amount (on the x-axis) and Kenny's final amount (on the y-axis). 

Jack's original amount is $162, so his final amount is $12.

This is much easier with a little algebra though!!