If the average of the lowest 15 scores is 15% less than the average of the 15 highest scores, then set the average of the highest 15 scores equal to x, making the average of the 15 lowest scores 0.85x. Use the Average Formula: [IMAGE: Average = Sum/# of Terms], which is also Sum = Average × # of Terms.
From the given average of all 30 scores: Sum of all 30 scores = 74 × 30
From the averages for the bottom 15 and top 15 scores: Sum of all 30 scores = (0.85x)(15) + (x)(15)
Putting the two ways of expressing the sum of all 30 scores together:
74 × 30 = (0.85x)(15) + (x)(15)
74 × 30 = 15[0.85x + x]
74 × 2 = 0.85x + x
148 = 1.85x
x = 148 ÷ 1.85
x = 80
Remember that x is the average of the highest 15 scores, and the question asks for the average of the lowest 15 scores, which is 15% less: compute either as 0.85(80) = 68 or 80 – 8 – 4 = 68. It makes sense that 68 and 80 are the averages of the bottom and top half of the scores, respectively, as these numbers average to 74.
The correct answer is (A).
Alternatively, work backwards from the answer choices.

average of the lowest 15 scores: 0.85x

average of the highest 15 scores:
x = answer choice/0.85

Average of other two columns

(A)

68

68/0.85 = 80

(68 + 80)/2 = 74

(B)

69

69/0.85 = 81.17+

approx (69 + 81.17)/2 = 75.085

(C)

74

74/0.85 = 87.05+

too high

(D)

79

79/0.85 = 92.94+

too high

(E)

80

80/0.85 = 94.11+

too high

Note that if you read the constraints and question carefully, you can eliminate (C), (D), and (E) with no calculations. If the average of all 30 scores is 74, and the 15 lowest scores average to less than the 15 highest scores do, then the average of the 15 lowest scores must be less than 74 while the average of the 15 highest scores must be greater than 74.
The correct answer is (A).