Question

If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a},$ then:

$a$ must equal $c$

$a+b+c+d$ must equal zero

either $a=c$ or $a+b+c+d=0,$ or both

$a+b+c+d\ne0$ if $a=c$

$a(b+c+d)=c(a+b+d)$

Answer: C

Concepts:rational equation factoring zero product property

Difficulty rating: 1450

Small Hint:

Cross-multiply the two fractions

Big Hint:

Move all terms to one side and factor out $a-c$

Solution:

Cross-multiplication gives $(a+b)(a+d)=(b+c)(c+d).$ Subtracting the right side and factoring yields $(a-c)(a+b+c+d)=0.$ Hence either $a=c$ or the sum is zero, and both may occur.

Therefore, the correct answer is C.

1964 AMC 12 Problem 32

Answer: C

Difficulty rating: 1450

Small Hint:

Big Hint:

Solution:

Problem 32 in Other Years