For a certain set, the value range of its members is 112.4. A new set is created from the members of the old set as follows: 12 is subtracted from a member of the old set, and the result is divided by 4. The resulting value is a member of the new set. If this operation is done for each member of the old set, what is the range of values of the members of the new set?
A. 25.1 B. 28.1 C. 56.4 D. 98.4 E. 112.4
(B) A well-practiced student’s intuition should tell us that the new range will be 112.4/4 = 28.1. But to be rigorous about it: Under the assumption that the range after the transformation will be the same for any set that has originally the range of 112.4, let the largest member of the old set be 0 and the smallest member be -112.4 . In the new set, the largest value will be (0 – 12)/4 and the smallest value will be (-112.4 – 12)/4. Thus, the range of the new set will be: (-12/4) – (-124.4 / 4) = -3 – (-31.1) = 31.1 – 3 = 28.1 . The correct answer is B.
The alternative method is the following: Suppose the greatest and the least of the elements of the original set are G and s, thus giving the range = G – s = 112.4 . Now after the transformation of subtracting 12 from each member and dividing the result by 4, the greatest and the least elements of the original set will be transformed into the greatest and the least elements of the new set. The range works out to be (G – 12)/4 – (s – 12)/4 which can be simplified: (G – s)/4 = 112.4 / 4 = 28.1 . The correct answer is B.
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