smc

Question

Suppose that S is a finite set of positive integers. If the greatest integer in S is removed from S, then the average value (arithmetic mean) of the integers remaining is 32. If the least integer in S is also removed, then the average value of the integers remaining is 35. If the greatest integer is then returned to the set, the average value of the integers rises to 40. The greatest integer in the original set S is 72 greater than the least integer in S. What is the average value of all the integers in the set S?

Accepted Answer

Let the lowest value be L and the highest G, and let the sum be Z and the amount of numbers n. We have Z-Gn-1=32, Z-L-Gn-2=35, Z-Ln-1=40, and G=L+72. Clearing denominators gives Z-G=32n-32, Z-L-G=35n-70, and Z-L=40n-40. We use G=L+72 to turn the first equation into Z-L=32n+40. Since Z-L=40(n-1) we substitute it into the equation which gives n=10. Turning the second into Z-2L=35n+2 using G=L+72 we see L=8 and Z=368 so the average is Zn=36.8