The question is:

"The sum of two numbers is 15. If the sum of their reciprocals is $\frac{9}{20}$, find the numbers."

Solution

Let the two numbers be $x$ and $y$.

1. Step 1: Given equations

   • Sum of the two numbers:
     $x + y = 15 \quad \text{(Equation 1)}$
   • Sum of their reciprocals:
     $\frac{1}{x} + \frac{1}{y} = \frac{9}{20} \quad \text{(Equation 2)}$

2. Step 2: Express the sum of reciprocals
   The left-hand side of Equation 2 can be written as:
   $\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}.$ Substituting $x + y = 15$ into this expression:
   $\frac{15}{xy} = \frac{9}{20}.$

3. Step 3: Solve for $xy$ Cross-multiply to find $xy$:
   $15 \times 20 = 9 \times xy \quad \Rightarrow \quad 300 = 9xy.$ Therefore,
   $xy = \frac{300}{9} = \frac{100}{3}.$

4. Step 4: Use a quadratic equation
   The numbers $x$ and $y$ are the roots of the quadratic equation:
   $t^2 - (x + y)t + xy = 0.$ Substituting $x + y = 15$ and $xy = \frac{100}{3}$, the equation becomes:
   $t^2 - 15t + \frac{100}{3} = 0.$ Multiply through by 3 to eliminate the fraction:
   $3t^2 - 45t + 100 = 0.$

5. Step 5: Solve the quadratic equation
   Use the quadratic formula:
   $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 3$, $b = -45$, and $c = 100$.

   First, calculate the discriminant:
   $b^2 - 4ac = (-45)^2 - 4 \times 3 \times 100 = 2025 - 1200 = 825.$

   Therefore,
   $t = \frac{45 \pm \sqrt{825}}{6}.$

6. Step 6: Approximate the roots
   $\sqrt{825} \approx 28.72.$ So,
   $t = \frac{45 \pm 28.72}{6}.$

   • One root:
     $t_1 = \frac{45 + 28.72}{6} \approx \frac{73.72}{6} \approx 12.29.$
   • The other root:
     $t_2 = \frac{45 - 28.72}{6} \approx \frac{16.28}{6} \approx 2.71.$

Approximate Solution

The two numbers are approximately 12.29 and 2.71.

Would you like a more detailed explanation, additional examples, or exact decimal values?

Follow-Up Questions

1. How can you use Vieta’s formulas to check the solution?
2. How do reciprocal equations relate to harmonic means?
3. What if the sum of the numbers was different? How would that change the steps?
4. What is the geometric interpretation of solving for roots in quadratic equations?
5. Could we apply other techniques like completing the square here?

Tip

In problems involving sum and reciprocal sum, combining equations into a quadratic form is the key step!