Question 1176896
my interpretation of this problem is:


A and B had some cards.
After A lost 68 cards to B, the ratio of cards that A had to B was 3/8.
When B subsequently lost 126 cards to A, the ratio of cards that A had to B became 2/3.



the first part of this problem became:


(A - 68) / (B + 68) = 3/8


the second part of this problem became:


(A + 58) / (B - 58) = 2/3.


the reason for this is that A first lost 68 cards and then won 126 cards, giving him a net gain of 58 cards and that B first won 68 cards and then lost 126 cards, giving him a net loss of 58 cards.


you have two equations that need to be solved simultaneously.


they are:


(A - 68) / (B + 68) = 3/8
(A + 58) / (B - 58) = 2/3


in the first equation, you get:
(A - 68) = 3/8 * (B + 68)
solve for A to get:
A = 3/8 * (B + 68) + 68.


in the second equation, you get:
(A + 58) = 2/3 * (B - 58)
solve for A to get:
A = 2/3 * (B - 58) - 58


since they are both equal to A, you get:


3/8 * (B + 68) + 68 = 2/3 * (B - 58) - 58


subtract 3/8 * (B + 68) from both sides of the equation and add 58 to both sides of the equation to get:


68 + 58 = 2/3 * (B - 58) - 3/8 * (B + 68)



simplify to get:


126 = 2/3 * B - 2/3 * 58 - 3/8 * B - 3/8 * 68


add 2/3 * 58 and 3/8 * 68 to both sides of the equation to get:


126 + 2/3 * 58 + 3/8 * 68 = 2/3 * B - 3/8 * B


put everything under the common denominator of 24 to get:


3024 / 24 + 928 / 24 + 612 / 24 = 16/24 * b - 9/24 * B


combine like terms to get:


4564 / 24 = 7/24 * B


solve for B to get:


B = 24/7 * 4564 / 24 = 652.


since A = 3/8 * (B + 68) + 68, then:


A = 3/8 * (652 + 68) + 68 = 338.


when A = 338 and B = 652:


(A - 68) / B + 68) becomes (338 - 68) / (652 + 68) = 270 / 720 = 3/8.


(A + 58) / (B - 58) becomes (338 + 58) / (652 - 58) = 396 / 594 = 2/3.


the requirements of the problem are satisfied.


your answer is that A had 338 cards at first.