Question

Sum of two numbers is 8 and sum of their inverses is \( \frac{8}{15} \). Find the numbers.

Hint:

When you know the sum and product of two numbers, forming a quadratic equation \(t^2 - (\text{sum})t + (\text{product}) = 0\) is the most direct way to find the numbers.

Answer 1

Step 1: Understanding the Concept:
We need to solve a system of two equations with two variables. One equation is linear, and the other involves reciprocals, which can be transformed into an equation involving the product of the variables.
Step 2: Key Formula or Approach:
Let the two numbers be \(x\) and \(y\). We are given:
1. \( x + y = 8 \)
2. \( \frac{1}{x} + \frac{1}{y} = \frac{8}{15} \)
We can solve this system to find \(x\) and \(y\).
Step 3: Detailed Explanation:
First, simplify the second equation:
\[ \frac{1}{x} + \frac{1}{y} = \frac{y+x}{xy} = \frac{x+y}{xy} \] So, the equation becomes:
\[ \frac{x+y}{xy} = \frac{8}{15} \] We are given from the first equation that \( x + y = 8 \). Substitute this into the simplified second equation:
\[ \frac{8}{xy} = \frac{8}{15} \] From this, we can deduce that \( xy = 15 \).
Now we have a simple system of equations:
(i) \( x + y = 8 \)
(ii) \( xy = 15 \)
We can solve this by substitution or by forming a quadratic equation. The quadratic equation with roots \(x\) and \(y\) is given by \( t^2 - (\text{sum of roots})t + (\text{product of roots}) = 0 \).
\[ t^2 - (x+y)t + xy = 0 \] \[ t^2 - 8t + 15 = 0 \] Factor the quadratic equation:
\[ (t - 3)(t - 5) = 0 \] The roots are \(t = 3\) and \(t = 5\).
Step 4: Final Answer:
The two numbers are 3 and 5.