To get a human or AI tutor to help you, click Register
Sample 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.
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?
\(\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66\)
Answer Keys
Question 13: BSolutions
Question 13Step 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.