Question 1184088
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Juliet gave her 1/5 of her marbles to Ken. After that, Ken gave 2/3 of what he had and an additional 4 more marbles to Juliet. Then, Juliet lost 29 marbles on the way home. In the end, Juliet had 97 marbles and Ken had 21 marbles left. How many marbles did each of them have at first?
<pre>Let original amounts Juliet and Ken had, be J, and K, respectively
After giving Ken {{{1/5}}} of her marbles, Juliet had {{{matrix(1,5, (4/5), of, "J,", or, 4J/5)}}} remaining 
After receiving {{{1/5}}} of Juliet's marbles, Ken then had: {{{matrix(1,5, 
(1/5), of, J + K, "=", J/5 + K)}}}  
After giving {{{2/3}}} of what he had and an additional 4 more marbles to Juliet, Ken then had: {{{matrix(1,5, (1/3), of, (J/5 + K) - 4, "or", J/15 + K/3 - 4)}}} remaining
Since Ken had a final count of 21 marbles, we then get: {{{system(matrix(1,3, J/15 + K/3 - 4, "=", 21), matrix(1,3, J + 5K - 60, "=", 315), matrix(1,6, J + 5K, "=", 375, "-------", eq, "(i)"))}}}  

After receiving {{{2/3}}} and an additional 4 more marbles from Ken, Juliet then had: {{{matrix(1,5, 4J/5 + (2/3), of, (J/5 + K) + 4, "or", 4J/5 + 2J/15 + 2K/3 + 4)}}} remaining
Since Juliet lost 29 marbles and ended up with 97 marbles, we then get: {{{system(matrix(1,3, 4J/5 + 2J/15 + 2K/3 + 4 - 29, "=", 97), matrix(1,3, 4J/5 + 2J/15 + 2K/3 - 25, "=", 97), matrix(1,3, 4J/5 + 2J/15 + 2K/3, "=", 122), matrix(1,3, 12J + 2J + 10K, "=", "1,830"), matrix(1,6, 14J + 10K, "=", "1,830", "-------", eq, "(ii)"))}}}
  J +  5K = 375 ------ eq (i)
14J + 10K = 1,830 ---- eq (ii)
 2J + 10K = 750 ------ Multiplying eq (i) by 2 ------- eq (iii) 
      12J = 1,080 ---- Subtracting eq (iii) from eq (ii)
Original number of marbles Juliet had, or {{{highlight_green(matrix(1,5, J, "=", "1,080"/12, "=", 90))}}} 

90 + 5K = 375 ------- Substituting 90 for J in eq (i)
     5K = 375 - 90
     5K = 285
Original number of marbles Ken had, or {{{highlight_green(matrix(1,5, K, "=", 285/5, "=", 57))}}}</pre>