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Sample Question 6:
Step 1: Let the two numbers be \(x\) and \(y\).
Step 2: Based on the prompt, form two equations: The first equation is \(xy = 9\) and the second equation is \(\frac{1}{x} = 4(\frac{1}{y})\).
Step 3: From the second equation, derive that \(4x = y\).
Step 4: Replace \(y\) with \(4x\) in the first equation to get \(4x^2 = 9\).
Step 5: Solve this equation to find \(x = \frac{3}{2}\).
Step 6: Substitute \(x = \frac{3}{2}\) into \(4x = y\) to find \(y = 6\).
Step 7: The sum of the two numbers is \(x + y = \frac{3}{2} + 6 = \frac{15}{2}\), which leads to answer \(\textbf{(D)}\ \frac{15}{2}\).
The product of two positive numbers is \(9\). The reciprocal of one of these numbers is \(4\) times the reciprocal of the other number. What is the sum of the two numbers?
\(\textbf{(A)}\ \frac{10}{3}\qquad\textbf{(B)}\ \frac{20}{3}\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ \frac{15}{2}\qquad\textbf{(E)}\ 8\)
Answer Keys
Question 6: DSolutions
Question 6Step 1: Let the two numbers be \(x\) and \(y\).
Step 2: Based on the prompt, form two equations: The first equation is \(xy = 9\) and the second equation is \(\frac{1}{x} = 4(\frac{1}{y})\).
Step 3: From the second equation, derive that \(4x = y\).
Step 4: Replace \(y\) with \(4x\) in the first equation to get \(4x^2 = 9\).
Step 5: Solve this equation to find \(x = \frac{3}{2}\).
Step 6: Substitute \(x = \frac{3}{2}\) into \(4x = y\) to find \(y = 6\).
Step 7: The sum of the two numbers is \(x + y = \frac{3}{2} + 6 = \frac{15}{2}\), which leads to answer \(\textbf{(D)}\ \frac{15}{2}\).