earnit wrote:
8 schools sent a total of 96 students to a math club competition, with each school sending at least one student. If one school sent the third most number of students to the competition (and no two schools sent the same number of students), did that school send at least 15 students?
(1) The most number of students that were sent by any one school was 34.
(2) The second most number of students that were sent by any school was 33.
We are given that 8 schools sent a total of 96 students to a math club competition, with each school sending at least one student. We also know that no two schools sent the same number of students. We need to determine whether the school that sent the third most number of students sent at least 15.
Statement One Alone:
The most number of students that were sent by any one school was 34.
The information in statement one is not sufficient to answer the question. For example, we can have the following scenarios of students sent, from greatest to least:
34 + 30 + 15 + 9 + 4 + 3 + 1 = 96
or
34 + 30 + 14 + 10 + 4 + 3 + 1 = 96
We see that the school that sent the third most students could have sent at least 15 students or fewer than 15 students.
Statement Two Alone:
The second most number of students that were sent by any school was 33.
The information in statement two is sufficient to answer the question. Let’s say the greatest number of students sent by a school was 34, the smallest number of students sent by a school was 1, the second smallest was 2, the third smallest was 3, the fourth smallest was 4, and the fifth smallest was 5, while the third most (the one in the equation) is x. We can create the following equation:
34 + 33 + x + 5 + 4 + 3 + 2 + 1 = 96
x + 82 = 96
x = 14
We see that, in this case, the third greatest is 14 students, which is less than 15. In the equation above, we can modify the numbers (except 33). However, any numbers we modify have to be larger; for example, if the greatest number of students is not 34, then it has to be 35 or larger. This will cause the value of x to be even smaller (i.e., less than 14). Thus, we will never have at least 15 students sent by the school with the third greatest number of students.
Answer: B
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