For the two positive numbers `a,b`,if `a,b` and `(1)/(18)` are in a geometric progression,while `(1)/(a),10` and `(1)/(b)` are in an arithmetic progression,then `16a+12b` is equal to

Answer

To solve the problem step by step, we need to analyze the conditions given in the question.

### Step 1: Set up the geometric progression condition
We know that \( a, b, \frac{1}{18} \) are in a geometric progression (GP). For three numbers \( A, B, C \) to be in GP, the condition is:
\[ B^2 = A \cdot C \]
In our case, this translates to:
\[ b^2 = a \cdot \frac{1}{18} \]
Rearranging this gives:
\[ b^2 = \frac{a}{18} \quad \text{(1)} \]

### Step 2: Set up the arithmetic progression condition
Next, we know that \( \frac{1}{a}, 10, \frac{1}{b} \) are in an arithmetic progression (AP). For three numbers \( A, B, C \) to be in AP, the condition is:
\[ 2B = A + C \]
In our case, this translates to:
\[ 2 \cdot 10 = \frac{1}{a} + \frac{1}{b} \]
Simplifying gives:
\[ 20 = \frac{1}{a} + \frac{1}{b} \quad \text{(2)} \]

### Step 3: Substitute and solve
From equation (1), we can express \( a \) in terms of \( b \):
\[ a = 18b^2 \]
Now, substitute this expression for \( a \) into equation (2):
\[ 20 = \frac{1}{18b^2} + \frac{1}{b} \]
To combine the fractions, we find a common denominator:
\[ 20 = \frac{1 + 18b}{18b^2} \]
Multiplying both sides by \( 18b^2 \) gives:
\[ 360b^2 = 1 + 18b \]
Rearranging this leads to:
\[ 360b^2 - 18b - 1 = 0 \quad \text{(3)} \]

### Step 4: Solve the quadratic equation
Now we will solve the quadratic equation (3) using the quadratic formula:
\[ b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \( A = 360, B = -18, C = -1 \):
\[ b = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 360 \cdot (-1)}}{2 \cdot 360} \]
Calculating the discriminant:
\[ b = \frac{18 \pm \sqrt{324 + 1440}}{720} \]
\[ b = \frac{18 \pm \sqrt{1764}}{720} \]
\[ b = \frac{18 \pm 42}{720} \]
Calculating the two possible values for \( b \):
1. \( b = \frac{60}{720} = \frac{1}{12} \)
2. \( b = \frac{-24}{720} \) (not valid since \( b \) must be positive)

Thus, we have:
\[ b = \frac{1}{12} \]

### Step 5: Find \( a \)
Now substitute \( b \) back into the expression for \( a \):
\[ a = 18b^2 = 18 \left(\frac{1}{12}\right)^2 = 18 \cdot \frac{1}{144} = \frac{18}{144} = \frac{1}{8} \]

### Step 6: Calculate \( 16a + 12b \)
Now we can calculate \( 16a + 12b \):
\[ 16a + 12b = 16 \cdot \frac{1}{8} + 12 \cdot \frac{1}{12} \]
Calculating each term:
\[ = 2 + 1 = 3 \]

### Final Answer
Thus, the value of \( 16a + 12b \) is:
\[ \boxed{3} \]