SAT problem of the week, October 27, 2014 solution

Hint for process of elimination: You need to know that “integers” are the positive and negative whole numbers, “nonnegative” means the numbers are not negative, and “distinct” means the list does not repeat any numbers. If the arithmetic mean of the five numbers is 7, then either all five numbers are 7, or at least one of the numbers is greater than 7. This helps you to eliminate two choices.

Another hint: They want to know the maximum value of the largest number is. The maximum value of the largest number occurs when the other four numbers are as small as possible. So instead of searching for the largest number, choose the other four numbers as small as possible, and use that to get the largest number, or at least an idea of how large the largest number could be.

Read on for a full solution.

Answer: D

Solution: “Distinct numbers” means the numbers are all different. If the smallest four numbers are as small as possible, then the largest numbers are as large as possible. The four smallest numbers can be 0, 1, 2, and 3. Let x be the largest number. We can then write down the arithmetic mean of the five numbers, and solve the equation:

7 = \dfrac{0+1+2+3+x}{5} = \dfrac{6+x}{5}

7 \times 5 = \dfrac{6+x}{\not{5}} \times \not{5}

35 - 6 = \not{6} + x - \not{6}

29 = x