1977 AMC 12 Problem 12

Attempt Problem 12 of the 1977 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1977 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

Al’s age is

16

more than the sum of Bob’s age and Carl’s age, and the square of Al’s age is

1632

more than the square of the sum of Bob’s age and Carl’s age. The sum of the ages of Al, Bob and Carl is

64

94

96

102

140

Let

s

be the sum of Bob’s and Carl’s ages

Factor

a^2-s^2

and use the known value of

a-s

Let

a

be Al’s age and

s

the sum of the other two ages. Then

a-s=16

and

\begin{aligned} 1632&=a^2-s^2\\ &=(a-s)(a+s)\\ &=16(a+s). \end{aligned}

Hence the requested total

a+s

\frac{1632}{16}=102.

Therefore, the correct answer is D.

Problem 12 in other years