When resistors 1 and 2 are connected in series, the equivalent resistance is `20.0 Omega.` When they are connected in parallel, the equivalent resistance is `3.75 Omega.` What are (a) the smaller resistance and (b) the larger resistance of these two resistors?

Answer

To solve the problem, we need to find the values of two resistors, R1 and R2, given their equivalent resistances in series and parallel configurations.

### Step-by-Step Solution:

1. **Set Up the Equations:**
- When resistors R1 and R2 are connected in series, the equivalent resistance (R_series) is given by:
\[ R_{series} = R_1 + R_2 = 20.0 \, \Omega \quad \text{(1)} \]
- When R1 and R2 are connected in parallel, the equivalent resistance (R_parallel) is given by:
\[ R_{parallel} = \frac{R_1 \cdot R_2}{R_1 + R_2} = 3.75 \, \Omega \quad \text{(2)} \]

2. **Express R1 in terms of R2:**
- From equation (1), we can express R1 as:
\[ R_1 = 20 - R_2 \quad \text{(3)} \]

3. **Substitute R1 in the Parallel Resistance Equation:**
- Substitute equation (3) into equation (2):
\[ 3.75 = \frac{(20 - R_2) \cdot R_2}{20} \]

4. **Multiply both sides by 20 to eliminate the fraction:**
\[ 75 = (20 - R_2) \cdot R_2 \]

5. **Expand and rearrange the equation:**
\[ 75 = 20R_2 - R_2^2 \]
Rearranging gives:
\[ R_2^2 - 20R_2 + 75 = 0 \quad \text{(4)} \]

6. **Solve the Quadratic Equation:**
- We can use the quadratic formula:
\[ R_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \(a = 1\), \(b = -20\), and \(c = 75\):
\[ R_2 = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 75}}{2 \cdot 1} \]
\[ R_2 = \frac{20 \pm \sqrt{400 - 300}}{2} \]
\[ R_2 = \frac{20 \pm \sqrt{100}}{2} \]
\[ R_2 = \frac{20 \pm 10}{2} \]

7. **Calculate the Two Possible Values for R2:**
- This gives us two solutions:
\[ R_2 = \frac{30}{2} = 15 \, \Omega \quad \text{(5)} \]
\[ R_2 = \frac{10}{2} = 5 \, \Omega \quad \text{(6)} \]

8. **Determine R1 Using R2 Values:**
- If \(R_2 = 15 \, \Omega\), then from equation (3):
\[ R_1 = 20 - 15 = 5 \, \Omega \]
- If \(R_2 = 5 \, \Omega\), then:
\[ R_1 = 20 - 5 = 15 \, \Omega \]

9. **Identify the Smaller and Larger Resistance:**
- The smaller resistance is \(5 \, \Omega\) and the larger resistance is \(15 \, \Omega\).

### Final Answers:
- (a) The smaller resistance is **5 Ω**.
- (b) The larger resistance is **15 Ω**.