Question 1164370

Jane and Tom had some sweets each. Jane  gave 1/2 of what she had to Tom. Then Tom gave 1/3 of what he had to Jane. Jane then gave 1/2 of what she had to Tom. As a result Jane  had 54 sweets and Tom had150 sweet. How many sweets did Tom have at first?
<pre>Let number that Tom and Jane had, at 1st be T, and J, respectively
After giving ½ to Tom, Jane had ½J, or {{{J/2}}} left 
After getting ½ from Jane, Tom had {{{T + J/2}}}


After giving {{{1/3}}} to Jane, Tom had {{{(2/3) * (T + J/2)}}}, or {{{2T/3 + J/3}}} left 
After getting {{{1/3}}} from Tom, Jane had {{{(1/3) * (T + J/2) + J/2}}}, or {{{T/3 + J/6 + J/2}}}, or {{{matrix(1,5, T/3 + J/6 + 3J/6, "=", T/3 + 4J/6, "=", T/3 + 2J/3)}}}, left


After giving ½ to Tom, Jane had ½ * {{{T/3 + 2J/3}}}, or {{{T/6 + J/3}}}, left 
After getting ½ from Jane, Tom had {{{matrix(1,7, T/6 + J/3 + 2T/3 + J/3, "=", T/6 + 2T/3 + 2J/3, "=", T/6 + 4T/6 + 2J/3, "=", 5T/6 + 2J/3)}}}

We now have: {{{matrix(1,3, 5T/6 + 2J/3, "=", 150)}}}
             5T + 4J = 900 ---- Multiplying by LCD, 6 ------- eq (i)
         Also, {{{matrix(1,3, T/6 + J/3, "=", 54)}}}
              T + 2J = 324 ---- Multiplying by LCD, 6 ------- eq (ii)
           - 2T - 4J = - 648 -- Multiplying eq (ii) by - 2 -- eq (iii)
                  3T = 252 ---- Adding eqs (iii) & (i)
Number of sweets Tom had, at first, or {{{highlight_green(matrix(1,5, T, "=", 252/3, "=", 84))}}}</pre>