Sample Question 13:

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$

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Answer Keys

Question 13: B

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Solutions

Question 13
Step 1: Understand that the total number of games must be the sum of games won by left- and right-handed players, hence the equation $g = l + r$.

Step 2: Since it is mentioned that the number of games won by left-handed players was 40% more than the number of games won by right-handed players, it leads to the equation $l = 1.4r$.

Step 3: Substitute the value of $l$ from the second equation into the first equation to get $g = 2.4r$.

Step 4: Since $r$ and $g$ are integers, their ratio $g/2.4$ must also be an integer. Thus, $g$ must be a multiple of 12, leaving only options B and D.

Step 5: Recall that the number of games in a tournament where every participant plays every other participant exactly once is given by the formula $n(n-1)/2$, the sum of the first $n-1$ integers.

Step 6: Setting 36 and 48 equal to the formula $n(n-1)/2$, we need to find two consecutive numbers that have a product of 72 or 96.

Step 7: Realize that 72 is the product of 8 and 9, which are consecutive numbers, hence option B is the correct answer; the total number of games played is 36.