1951 AHSME Problems/Problem 17

Problem

Indicate in which one of the following equations is neither directly nor inversely proportional to :

$\textbf{(A)}\ x \plus{} y \equal{} 0 \qquad\textbf{(B)}\ 3xy \equal{} 10 \qquad\textbf{(C)}\ x \equal{} 5y \qquad\textbf{(D)}\ 3x \plus{} y \equal{} 10$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(E)}\ \frac {x}{y} \equal{} \sqrt {3}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Notice that for any directly or inversely proportional values, it can be expressed as or . Now we try to convert each into its standard form counterpart. $\textbf{(A)}\ x \plus{} y \equal{} 0\implies \frac{x}{y}=-1$ (Error compiling LaTeX. Unknown error_msg)

$\textbf{(B)}\ 3xy \equal{} 10\implies xy=\frac{10}{3}$ (Error compiling LaTeX. Unknown error_msg)

$\textbf{(C)}\ x \equal{} 5y\implies \frac{x}{y}=5$ (Error compiling LaTeX. Unknown error_msg)

$\textbf{(E)}\ \frac {x}{y} \equal{} \sqrt {3}\implies \frac {x}{y} \equal{} \sqrt {3}$ (Error compiling LaTeX. Unknown error_msg)

As we can see, the only equation without a "standard" form is , so our answer is $\boxed{\textbf{(D)}\ 3x \plus{} y \equal{} 10}$ (Error compiling LaTeX. Unknown error_msg)