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2021 AMC 12B Problem 12

Problem 12 of 25IntermediateAlgebraProbability & Statistics

Suppose that SS is a finite set of positive integers. If the greatest integer in SS is removed from S,S, then the average value (arithmetic mean) of the integers remaining is 32.32. If the least integer in SS is also removed, then the average value of the integers remaining is 35.35. If the greatest integer is then returned to the set, the average value of the integers rises to 40.40. The greatest integer in the original set SS is 7272 greater than the least integer in S.S. What is the average value of all the integers in the set S?S?

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Solution

Let n=∣S∣,n=|S|, let TT be the total, MM the greatest, and LL the least. Then T−Mn−1=32,\dfrac{T-M}{n-1}=32, T−M−Ln−2=35,\dfrac{T-M-L}{n-2}=35, and T−Ln−1=40.\dfrac{T-L}{n-1}=40. Subtracting the first from the third: M−Ln−1=8.\dfrac{M-L}{n-1}=8. Since M−L=72,M-L=72, we get n−1=9,n-1=9, so n=10.n=10. Then T−M=288T-M=288 and T−L=360.T-L=360. The middle equation gives T−M−L=35⋅8=280,T-M-L=35\cdot 8=280, so L=288−280=8L=288-280=8 and M=80.M=80. Thus T=288+80=368,T=288+80=368, and the average is 36810=36.8.\dfrac{368}{10}=36.8. Thus, the correct answer is D.
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Tagged: mean · system of equations

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