The simplest form of 1-11+(a)/(1-a) is:

(D) 1-a with no restriction on a

1957 AMC 12 Problem 3

Attempt Problem 3 of the 1957 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 1957 AMC 12 solutions, or check the answer key.

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The simplest form of

1-\dfrac{1}{1+\dfrac{a}{1-a}}

is:

a

a\ne0

1

a

a\ne-1

1-a

with no restriction on

a

a

a\ne1

Combine

1+\frac{a}{1-a}

before taking its reciprocal

Retain every restriction imposed by the original denominator

The original expression requires

a\ne1.

For such

a,

\begin{aligned} 1+\frac{a}{1-a} &=\frac{1-a+a}{1-a}\\ &=\frac1{1-a}. \end{aligned}

Therefore the expression is

1-(1-a)=a.

The restriction

a\ne1

remains.

Thus, the correct answer is E.