The AM, HM and GM between two numbers 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 determine the values of the Harmonic Mean (HM), Geometric Mean (GM), and Arithmetic Mean (AM) from the given numbers: \( \frac{144}{15} \), \( 15 \), and \( 12 \).

### Step 1: Identify the Means
We know that:
- The Arithmetic Mean (AM) is the largest value among the three.
- The Geometric Mean (GM) is the middle value.
- The Harmonic Mean (HM) is the smallest value.

### Step 2: Convert \( \frac{144}{15} \) to Decimal
First, let's convert \( \frac{144}{15} \) into decimal form:
\[ \frac{144}{15} = 9.6 \]

### Step 3: List the Values
Now we have the three values:
- \( 9.6 \) (from \( \frac{144}{15} \))
- \( 15 \)
- \( 12 \)

### Step 4: Determine the Order
Now we can order these values:
- Largest: \( 15 \) (AM)
- Middle: \( 12 \) (GM)
- Smallest: \( 9.6 \) (HM)

### Step 5: Assign the Means
From the order we determined:
- **AM** = \( 15 \)
- **GM** = \( 12 \)
- **HM** = \( 9.6 \)

### Final Result
Thus, the values of HM, GM, and AM respectively are:
- HM = \( 9.6 \)
- GM = \( 12 \)
- AM = \( 15 \)