If the mean of 3, 4, 9, 2k, 10, 8, 6 and (k + 6) is 8, and mode of 2, 2, 3, 2p, (2p + 1), 4, 4, 5 and 6 (p is a natural number) is 4, then the value of (k - 2p) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
The mean is the sum of the values divided by how many there are, and the mode is the value that occurs most often. The mean gives k; the mode gives p.
Finding k (eight values with mean 8):
Sum of the values \(=3+4+9+2k+10+8+6+(k+6)=46+3k\)
Eight values with mean 8 have a sum of \(8\times8=64\), so \(46+3k=64\)
\(3k=18\), so \(k=6\)
Finding p (mode 4):
Without 2p and 2p + 1, the value 2 occurs twice and 4 occurs twice, so 4 can be the only mode only if one of 2p and 2p + 1 is also 4
2p + 1 is odd and cannot be 4, so \(2p=4\) and \(p=2\) (a natural number, as the question requires)
The values are then 2, 2, 3, 4, 5, 4, 4, 5, 6, in which 4 occurs three times
The value asked:
With \(k=6\) and \(p=2\): \(k-2p=6-4=2\)