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: Problem Definition:
Given two numbers, \( x \) and \( y \), satisfying the following:
- Their sum is 18: \( x + y = 18 \)
- The sum of their reciprocals is \( \frac{1}{4} \): \( \frac{1}{x} + \frac{1}{y} = \frac{1}{4} \}

Step 2: Equation Simplification:
Combine the terms in the reciprocal equation: \( \frac{x + y}{xy} = \frac{1}{4} \).
Substitute \( x + y = 18 \): \( \frac{18}{xy} = \frac{1}{4} \).
Solve for the product \( xy \): \( xy = 18 \times 4 = 72 \).

Step 3: System of Equations:
We have the system:
1. \( x + y = 18 \)
2. \( xy = 72 \)
This represents the sum and product of two numbers.

Step 4: Quadratic Equation Formulation:
A quadratic equation with roots \( x \) and \( y \) is \( t^2 - (x + y)t + xy = 0 \).
Substitute the known sum and product: \( t^2 - 18t + 72 = 0 \).

Step 5: Quadratic Equation Solution:
Solve \( t^2 - 18t + 72 = 0 \) using the quadratic formula: \( t = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(1)(72)}}{2(1)} \).
Simplify: \( t = \frac{18 \pm \sqrt{324 - 288}}{2} = \frac{18 \pm \sqrt{36}}{2} = \frac{18 \pm 6}{2} \).
The possible values for \( t \) are: \( t = \frac{18 + 6}{2} = 12 \) or \( t = \frac{18 - 6}{2} = 6 \).

Step 6: Final Answer:
The two numbers are 12 and 6.