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Liam, Noah, and Olivia received their science project scores today. If the average (arithmetic mean) of their three scores was 85, what is the median of the three scores?
(1) Liam scored an 80 on his project.
(2) Olivia scored an 85 on her project.
(1) Liam scored an 80 on his project.
(2) Olivia scored an 85 on her project.
29 Apr 2025, 04:15
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Official Solution:
Liam, Noah, and Olivia received their science project scores today. If the average (arithmetic mean) of their three scores was 85, what is the median of the three scores?
(1) Liam scored an 80 on his project.
We can distribute the other two scores in different ways to get different medians. For example, {80, 80, 95} would give a median of 80, while {80, 85, 90} would give a median of 85. Not sufficient.
(2) Olivia scored an 85 on her project.
Olivia scored exactly the same as the average score, which implies that one of the other two must have scored less than or equal to 85 and the other greater than or equal to 85, making Olivia’s score the middle score, and thus the median equal to 85. That’s because if both of the other two had scored more than 85, the average would have been greater than 85; if both had scored less than 85, the average would have been less than 85. Therefore, we must have had the case {a ≤ (Olivia’s 85) ≤ b}. Sufficient.
Answer: B
Liam, Noah, and Olivia received their science project scores today. If the average (arithmetic mean) of their three scores was 85, what is the median of the three scores?
(1) Liam scored an 80 on his project.
We can distribute the other two scores in different ways to get different medians. For example, {80, 80, 95} would give a median of 80, while {80, 85, 90} would give a median of 85. Not sufficient.
(2) Olivia scored an 85 on her project.
Olivia scored exactly the same as the average score, which implies that one of the other two must have scored less than or equal to 85 and the other greater than or equal to 85, making Olivia’s score the middle score, and thus the median equal to 85. That’s because if both of the other two had scored more than 85, the average would have been greater than 85; if both had scored less than 85, the average would have been less than 85. Therefore, we must have had the case {a ≤ (Olivia’s 85) ≤ b}. Sufficient.
Answer: B
anshumannanda
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03 Nov 2025, 07:17
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I like the solution - it’s helpful.
