Question 1185489
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The problem is indeed very poorly stated.  It would be wrong if the statement of the problem in the textbook is exactly as you how it.<br>
And the textbook answer you show is wrong....<br>
Arthur and Sam shared some cards.
If Arthur gave 20 to Sam they would have the same number of cards. 
If Sam gave Arthur 10 cards the ratio of cards Arthur had to Sam {{{cross(is)}}} would be 2 to 1.
How many cards did Arthur have? {{{cross("at the end")}}}<br>
Indeed, as one of the tutors says, asking how many cards Arthur had "at the end" is misleading, because no actual exchanges of cards were made.<br>
For the same reason, the interpretation by the other tutor that the two exhanges actually took place one after the other is not right.<br>
So here is the way I see the problem -- apparently the same way your son saw it, because his answer is right.<br>
Let A = # of cards Arthur has
Let S = # of cards Sam has<br>
(1) A-20 = S+20  if Arthur gave Sam 20 cards, they would have the same number of cards<br>
(2) A+10=2(S-10)  if Sam gave Arthur 10 cards, Arthur would have twice as many cards as Sam<br>
There are many different ways to solve the pair of equations to find the answer to the problem.  I solved (2) for A and substituted in (1):<br>
A+10=2(S-10)
A+10=2S-20
A=2S-30
2S-30-20=S+20
S=70
A=2S-30=140-30=110<br>
ANSWER: Arthur had 110 cards<br>
CHECK:
Arthur started with 110 cards and Sam started with 70
If Arthur gave Sam 20 cards, they would both have the same number, 90
If Sam gave Arthur 10 cards, Arthur would have 120 and Sam would have 60; 120 is twice 60<br>
All the conditions are satisfied; our answer is correct<br>
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Another tutor, while I was writing my response, pointed out that Arthur would end up with 120 cards if the second described exchange -- and ONLY the second one -- actually took place.<br>
If that is the interpretation that is intended, so that the answer 120 is the intended answer, then the level of sloppiness in the presentation of the problem is even worse.  The statements regarding the two exchanges start with "If Arthur gave Sam... and "If Sam gave Arthur..." -- clearly indicating that NEITHER exchange actually took place.<br>
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I have seen your message that the problem you posted was taken exactly as given in your textbook....  Neither you nor I can do anything about that.<br>