The sum of two numbers is 15. If the sum of their reciprocals is $\frac{3}{10}$, then the larger number is

Question

Accepted Answer

To solve the problem, we need to find two numbers whose sum is 15 and the sum of their reciprocals is $\frac{3}{10}$. We are tasked with identifying the larger of these two numbers.

Let's denote the two numbers as $x$ and $y$, where $x > y$.

1. **Equation for the sum of the numbers:**
   $$
   x + y = 15
   $$

2. **Equation for the sum of their reciprocals:**
   $$
   \frac{1}{x} + \frac{1}{y} = \frac{3}{10}
   $$

This can be rewritten as:
   $$
   \frac{x + y}{xy} = \frac{3}{10}
   $$

Substituting $x + y = 15$ into the equation:
   $$
   \frac{15}{xy} = \frac{3}{10}
   $$

Solving for $xy$:
   $$
   15 \times 10 = 3 \times xy \implies xy = 50
   $$

3. **Solving the system of equations:**
   We have two equations:
   $$
   x + y = 15
   $$
   $$
   xy = 50
   $$

These can be solved using the quadratic equation:
   $$
   t^2 - (x+y)t + xy = 0
   $$
   Substituting the known values:
   $$
   t^2 - 15t + 50 = 0
   $$

Solving this quadratic equation using the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -15$, and $c = 50$:
   $$
   t = \frac{15 \pm \sqrt{15^2 - 4 \times 1 \times 50}}{2 \times 1}
   $$
   $$
   t = \frac{15 \pm \sqrt{225 - 200}}{2}
   $$
   $$
   t = \frac{15 \pm \sqrt{25}}{2}
   $$
   $$
   t = \frac{15 \pm 5}{2}
   $$

This gives us two solutions:
   $$
   t = \frac{20}{2} = 10 \quad \text{and} \quad t = \frac{10}{2} = 5
   $$

Therefore, the two numbers are 10 and 5. Since we are asked for the larger number, the answer is 10.

**Explanation of why other options are incorrect:**

- **Option 1 (15):** This cannot be correct because if one number were 15, the other would be 0 (since their sum is 15), which would make the reciprocal undefined.
- **Option 2 (20):** This is incorrect because if one number were 20, the other would be -5, which does not satisfy the condition for the sum of reciprocals.
- **Option 4 (25):** This is incorrect because if one number were 25, the other would be -10, which again does not satisfy the condition for the sum of reciprocals.

Thus, the correct answer is Option 3, which is 10.