The AM, HM and GM between two number are `(144)/(15)`, 15 and 12, but not necessarily in this order then, HM, GM and AM respectively are

Options

A. `(144)/(15),12, 15`
B. `(144)/(15), 15, 12`
C. `15, 12, (144)/(15)`
D. `12, 15, (144)/(15)`

Answer

To solve the problem, we need to identify the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) from the given values: \( \frac{144}{15} \), 15, and 12. We will determine which value corresponds to AM, GM, and HM based on their properties.

### Step-by-Step Solution:

1. **Understanding the Means**:
- The Arithmetic Mean (AM) of two numbers \( A \) and \( B \) is given by:
\[ AM = \frac{A + B}{2} \]
- The Geometric Mean (GM) is given by:
\[ GM = \sqrt{AB} \]
- The Harmonic Mean (HM) is given by:
\[ HM = \frac{2AB}{A + B} \]

2. **Properties of the Means**:
- We know that \( AM \geq GM \geq HM \) for any two positive numbers \( A \) and \( B \). This means that AM is the largest, GM is in the middle, and HM is the smallest.

3. **Identifying the Values**:
- We have three values: \( \frac{144}{15} \approx 9.6 \), 15, and 12.
- Let's compare these values:
- \( \frac{144}{15} \approx 9.6 \)
- 12
- 15
- From this comparison, we can see that:
\[ HM < GM < AM \]
Thus, \( \frac{144}{15} \) must be HM, 12 must be GM, and 15 must be AM.

4. **Conclusion**:
- Therefore, we can conclude:
- HM = \( \frac{144}{15} \)
- GM = 12
- AM = 15

### Final Answer:
- HM = \( \frac{144}{15} \)
- GM = 12
- AM = 15