SOLUTION: Hi A and B had some cards. After A lost 68 cards to B the ratio of cards that A and B had were 3:8 respectively. When B lost 126 cards to A the ratio of cards that A and B had we

Question 1176896: Hi
A and B had some cards. After A lost 68 cards to B the ratio of cards that A and B had were 3:8 respectively. When B lost 126 cards to A the ratio of cards that A and B had were 2:3 respectively.
How many cards did A have at first.
Thanks
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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.

Answer by MathTherapy(10556) (Show Source): You can put this solution on YOUR website!

Hi
A and B had some cards. After A lost 68 cards to B the ratio of cards that A and B had were 3:8 respectively. When B lost 126 cards to A the ratio of cards that A and B had were 2:3 respectively.
How many cards did A have at first.
Thanks

Let original number A and B had, be A and B, respectively
After A lost 68 cards to B, A had A - 68 remaining, and B then had, B + 68
We then get:
16A - 6B = 1,496 ---- Multiplying eq (i) by 2 ---- eq (iii)
9A - 6B = - 870 ---- Multiplying eq (ii) by 3 --- eq (iv)
7A = 2,366 -------- Subtracting eq (iv) from eq (iii)
Original number A had, or
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