Question

The sum of two numbers is 18, and the sum of their reciprocals is $\frac{1}{4}$. Find the numbers.

Answer 1

Step 1: Assign variables $x$ and $y$ to the two numbers. Establish the following equations: \[ x + y = 18 \tag{1}. \] \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{4} \tag{2}. \] Step 2: Reorganize equation (2). Employ the identity $\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}$: \[ \frac{x + y}{xy} = \frac{1}{4}. \] Substitute the value of $x + y$ from equation (1): \[ \frac{18}{xy} = \frac{1}{4} \implies xy = 72 \tag{3}. \] Step 3: Determine the roots of the quadratic equation. Based on equations (1) and (3), $x$ and $y$ are the roots of the quadratic equation: \[ t^2 - (x + y)t + xy = 0 \implies t^2 - 18t + 72 = 0. \] Factor the quadratic equation: \[ t^2 - 18t + 72 = (t - 12)(t - 6) = 0. \] This yields: \[ t = 12 \quad \text{or} \quad t = 6. \] Step 4: Identify the numbers. The two numbers are $12$ and $6$. Correct Answer: The numbers are $12$ and $6$.