1955 AHSME Problems/Problem 40

The fractions $\frac{ax+b}{cx+d}$ and $\frac{b}{d}$ are unequal if:

$\textbf{(A)}\ a=c=1, x\neq 0\qquad\textbf{(B)}\ a=b=0\qquad\textbf{(C)}\ a=c=0\\ \textbf{(D)}\ x=0\qquad\textbf{(E)}\ ad=bc$

Solution

We can implement each of these options on the existing two fractions: $\textbf{(A)}$ : $\frac{x+b}{x+d}$ vs $\frac{b}{d}$ . Since $x \neq 0$ , we can see that any remaining value of $x$ will make the two unequal. The answer is $\boxed{\textbf{(A)}}$

$\textbf{(B)}$ : $\frac{0}{d}$ vs $\frac{0}{d}$ Both fractions will equal zero $(d \neq 0)$ , so they are equivalent.

$\textbf{(C)}$ : $\frac{b}{d}$ vs $\frac{b}{d}$ The two fractions are equal.

$\textbf{(D)}$ : $\frac{b}{d}$ vs $\frac{b}{d}$ The two fractions are equal.

$\textbf{(E)}$ : This value will vary, but it doesn't take long to figure out a counterexample.

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1955 AHSME Problems/Problem 40

Solution

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