1970 AMC 12 Problem 35

Question

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 been

p

dollars greater, whereas, had he served

b

years more (

b\ne a

), his pension would have been

q

dollars greater than the original annual pension. Find his annual pension in terms of

a,

b,

p,

and

q.

\dfrac{p^2-q^2}{2(a-b)}

\dfrac{(p-q)^2}{2\sqrt{ab}}

\dfrac{ap^2-bq^2}{2(ap-bq)}

\dfrac{aq^2-bp^2}{2(bp-aq)}

\sqrt{(a-b)(p-q)}

Let the current pension be

X=k\sqrt n

and write the two hypothetical pensions

Square

X+p=k\sqrt{n+a}

and

X+q=k\sqrt{n+b}

, then eliminate both

n

and

k^2

Let the current pension be

X=k\sqrt n.

Since

X+p=k\sqrt{n+a}

and

X+q=k\sqrt{n+b},

squaring and using

X^2=k^2n

gives the first line below. Multiplying its equations by

b

and

a,

respectively, and subtracting gives the second:

\begin{gathered} 2pX+p^2=k^2a,\\ 2qX+q^2=k^2b,\\ 2X(bp-aq)=aq^2-bp^2. \end{gathered}

Hence

X=\dfrac{aq^2-bp^2}{2(bp-aq)}.

Therefore, the correct answer is D.

Problem 35 in other years