Solving min/max questions

Try the problem below: Five pieces of wood have an average length (arithmetic mean) of 124 centimeters and a median length of 140 centimeters. What is the maximum length, in centimeters, of the shortest piece of wood?  A) 90  B) 100  C) 110  D) 130  E) 140 First, let’s determine the size. If the average is 124 centimeters, and there are five pieces of wood, we know that the total sum of all the pieces of wood would be 5*124 = 620. Let’s call the smallest piece, ’s.’ So far, we have the following: s ___, 140, ___, ___ Next, we want to maximize the smallest piece. Think pie. If I want to maximize the size of one piece, in this case ‘s,’ I want to minimize the size of all the other slices. The minimum size for the second smallest slice is ‘s.’ (If it were any smaller, it would be the smallest slice.) The minimum size for our two larges slices is 140. (If those were any smaller, the median would change.) Now, we’re left with the following set: s, s, 140, 140, 140. Well, we already know that the sum is 620, so now we have the following equation: s + s + 140 + 140 + 140 = 620. 2s + 420 = 620 2s = 200 s = 100. The answer is B. Let’s try a tougher one: For a certain race, 3 teams were allowed to enter 3 members each. A team earned 6 – n points whenever one of its members finished in nth place, where 1 ≤ n ≤ 5. There were no ties, disqualifications, or withdrawals. If no team earned more than 6 points, what is the least possible score a team could have earned? A) 0 B) 1 C) 2 D) 3 E) 4 We know it’s a min/max question, so first we need to determine the size of the pie. We’re told that a team will earn 6 – n points whenever one of its members finishes in nth place. The team that has the first place finisher (n = 1) will earn 6 – 1 = 5 points. The second place finisher (n =2) will earn 6 – 2 = 4 points. The trend quickly becomes clear: First place: 5 points Second place: 4 points Third place: 3 points Fourth place: 2 points Fifth place: 1 point One of the conditions of the problem is that ‘n’ cannot be any larger than 5, so at this point, there are no more points to earn. Summing all the available points, we get 1 + 2 + 3 + 4 + 5 = 15. So there are 15 points total for the three teams to divvy up. Now we’re trying to minimize the number of points one team earned. We’re told that no team scored more than 6 points, so 6 is the max number of points a team could have earned. If two teams earned the max – 6 points – they’d have earned 12 points between them. If there are 15 points total, and two of the teams earn a total of 12 points, that leaves 3 points for the stragglers. D is the answer.