Question

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.

1955 AMC 12 Problem 47

Answer: C

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 47 in Other Years