1955 AMC 12 Problem 47

Attempt Problem 47 of the 1955 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 1955 AMC 12 solutions, or check the answer key.

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47.

The expressions

a+bc

and

(a+b)(a+c)

are:

always equal

never equal

equal when

a+b+c=1

equal when

a+b+c=0

equal only when

a=b=c=0

Answer: C

Concepts:algebraic expansion sufficient condition equality

Difficulty rating: 1280

Small hint:

Expand

(a+b)(a+c)

Big hint:

Subtract

a+bc

and factor the difference

Solution:

The difference is

\begin{aligned} &(a+b)(a+c)-(a+bc)\\ &\qquad=a^2+ab+ac-a\\ &\qquad=a(a+b+c-1). \end{aligned}

In particular, whenever

a+b+c=1,

this difference is zero and the expressions are equal.

Thus, the correct answer is C.

Problem 47 in other years