UNO High School Problem of the Week Competition
Problem 19
A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served A years more, his pension would have ben P dollars greater, whereas, had he served B years more (B ¹ A), his pension would have been Q dollars greater than the orignial annual pension. Find his annual pension in terms of A, B, P and Q.
Show your work!
Solution
Let X be the amount of the annual pension and y the number of years of service. Then with constant of proportionality k,
X = k√y
X2 = k2y
X + p = k√(y + a)
X + q = k√(y + b)
(x + p)2 = k2(y + a)
(X + q)2 = k2(y + b)
Replacing k2y by X2 in the last two equations and simplifying,
2pX + p2 = k2a and 2pX + q2 = k2b
Hence
(2pX + p2) / (2qX + q2) = a/b
Therefore, X = (aq2 - bp2) / 2(bp-aq).
