Devika has scored 62, 71, 78 and 85 in four tests and will write one more. What is the LOWEST score in the fifth test that makes the median of all five scores equal to 78?

With five scores the median is the 3rd one in order. The four known scores are 62, 71, 78, 85; if the new score $x$ is below 71 the order begins 62, 71 and the median is 71, while if 71 < $x$ < 78 the middle score is $x$ itself. Only $x$ ≥ 78 pushes 78 into the middle place — for example 62, 71, 78, 85, 90 has median 78 — so the lowest such score is 78 itself. The value 74.5 is the median of the four scores she already has, (71 + 78)/2, which the fifth test will replace. The value 94 answers the MEAN version of the question, since $5\times 78-296=94$; a mean has to be paid for exactly, while a median only has to be reached, which is why the two answers are so far apart. The score 71 leaves the median where it is now.