1957 AHSME Problems/Problem 8

The numbers $x,\,y,\,z$ are proportional to $2,\,3,\,5$ . The sum of $x, y$ , and $z$ is $100$ . The number y is given by the equation $y = ax - 10$ . Then a is:

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac{3}{2}\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ \frac{5}{2}\qquad \textbf{(E)}\ 4$

Solution

In order to solve the problem, we first need to find each of the three variables. We can use the proportions the problem gives us to find the value of one part, and, by extension, the values of the variables (as $x$ would have $2$ parts, $y$ would have $3$ , and $z$ would have $5$ ). One part, after some algebra, equals $10$ , so $x$ , $y$ , and $z$ are $20$ , $30$ , and $50$ , respectively.

We can plug $x$ and $y$ into the equation given to us: $30 = 20a-10$ , and then solve to get $a = \boxed{\textbf{(A)}2}$ .

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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1957 AHSME Problems/Problem 8

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