The arithmetic mean and geometric mean of two numbers are 10 and 18 respectively. What is the harmonic mean of the numbers

Options

A. `50/7`
B. `90/7`
C. `40/3`
D. `30/7`

Answer

To find the harmonic mean of two numbers given their arithmetic mean (AM) and geometric mean (GM), we can follow these steps:

### Step 1: Identify the given values
We are given:
- Arithmetic Mean (AM) = 10
- Geometric Mean (GM) = 18

### Step 2: Recall the formula for Harmonic Mean (HM)
The formula for the harmonic mean of two numbers \(a\) and \(b\) is given by:

\[ HM = \frac{2 \times AM \times GM}{AM + GM} \]

### Step 3: Substitute the values into the formula
Now, we can substitute the given values into the formula:

\[ HM = \frac{2 \times 10 \times 18}{10 + 18} \]

### Step 4: Calculate the numerator
First, calculate the numerator:

\[ 2 \times 10 \times 18 = 360 \]

### Step 5: Calculate the denominator
Next, calculate the denominator:

\[ 10 + 18 = 28 \]

### Step 6: Divide the numerator by the denominator
Now, we can compute the harmonic mean:

\[ HM = \frac{360}{28} \]

### Step 7: Simplify the fraction
To simplify \( \frac{360}{28} \), we can divide both the numerator and the denominator by 4:

\[ HM = \frac{90}{7} \]

### Conclusion
Thus, the harmonic mean of the two numbers is:

\[ HM = \frac{90}{7} \]