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.

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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

$64$

$94$

$96$

$102$

$140$

Answer: D

Concepts:difference of squares ages

Difficulty rating: 1360

Small Hint:

Let $s$ be the sum of Bob’s and Carl’s ages

Big Hint:

Factor $a^2-s^2$ and use the known value of $a-s$

Solution:

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 $a+s$ is $\frac{1632}{16}=102.$

Therefore, the correct answer is D.

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