1960 AMC 12 Problem 39

Question

To satisfy the equation

\frac{a+b}{a}=\frac{b}{a+b},

a

and

b

must be:

both rational

both real but not rational

both not real

one real, one not real

one real, one not real or both not real

Let

t=\frac{b}{a}

after noting that the denominators must be nonzero

The resulting quadratic

t^2+t+1=0

has negative discriminant

The denominators require

a\ne0

and

a+b\ne0.

Setting

t=\frac{b}{a}

and cross-multiplying gives

(1+t)^2=t,

or

t^2+t+1=0.

Its discriminant is

-3,

so

t

is not real. Therefore

a

and

b

cannot both be real. Depending on the nonzero complex scale chosen for

a,

one can be real and the other nonreal, or both can be nonreal.

Thus, the correct answer is E.

Problem 39 in other years