A university conducted an entrance test for a research program, with ten candidates participating. Each candidate was assigned a distinct integer score based on their performance. The top candidate scored 90 points. The difference in score between the top candidate and the seventh-ranked candidate was 25, while the difference in score between the fourth-ranked and tenth-ranked candidates was 6. Select for Median the median score of the ten participants, and for Range the range of the scores of the ten participants. Make only two selections, one in each column.

Official Solution: Given that the difference in score between the top candidate and the seventh-ranked candidate was 25, the 7th candidate scored 90 - 25 = 65 points. From the 4th to the 10th candidate, inclusive, there are 7 people: (4th, 5th, 6th, 7th, 8th, 9th, 10th). Since the range of their scores is 6 and all scores are distinct integers, their scores must be consecutive integers. With the 7th candidate scoring 65, the scores are: 4th = 68, 5th = 67, 6th = 66, 7th = 65, 8th = 64, 9th = 63, and 10th = 62. Therefore, the median score would be the average of the two middle scores, 5th and 6th, so 66.5, and the range would be the difference between the highest and lowest scores, so the difference between the top, 90, and the 10th, 62, so 28. Correct answer: Median "66.5" Range "28" GMAT-Club-Forum-mt7g4myx.png

1. Let's analyze the conditions that are given. The 7th candidate must've gotten 90-25 = 65 points. 1 2 3 4 5 6 7 8 9 10 90 ? ? ? ? ? 65 ? ? ? 2. All of the scores are distinct, so the score in 4th place is \geq 68 and the score in 10th place is \leq 62 . Their difference is 6, which is only possible if the scores were 68 and 62, respectively. 1 2 3 4 5 6 7 8 9 10 90 ? ? 68 ? ? 65 ? ? 62 3. This means that 5th, 6th, 8th, and 9th places must be consecutive numbers: 1 2 3 4 5 6 7 8 9 10 90 ? ? 68 67 66 65 64 63 62 4. The range of the series is 90 - 62 = 28 and the median is \frac{67 + 66}{2} = 66.5 . 5. Our answer will be: Median - 66.5 and Range - 28 .