If 256 is the third proportional of a and b, where a:b = 3:16, then find the value of a +b.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the concept of third proportional
If 256 is the third proportional of \( a \) and \( b \), it means that the ratio \( \frac{a}{b} \) is equal to \( \frac{b}{256} \).

### Step 2: Set up the ratio
Given that \( a:b = 3:16 \), we can express this as:
\[ \frac{a}{b} = \frac{3}{16} \]

### Step 3: Use the property of third proportional
From the property of third proportional, we have:
\[ \frac{a}{b} = \frac{b}{256} \]
This can be rearranged to:
\[ b^2 = 256a \]

### Step 4: Express \( a \) in terms of \( b \)
From the ratio \( \frac{a}{b} = \frac{3}{16} \), we can express \( a \) as:
\[ a = \frac{3}{16}b \]

### Step 5: Substitute \( a \) in the equation
Now substitute \( a \) in the equation \( b^2 = 256a \):
\[ b^2 = 256 \left(\frac{3}{16}b\right) \]
This simplifies to:
\[ b^2 = 256 \cdot \frac{3}{16} b \]

### Step 6: Simplify the equation
Calculating \( 256 \cdot \frac{3}{16} \):
\[ 256 \div 16 = 16 \quad \text{so} \quad 256 \cdot \frac{3}{16} = 48b \]
Thus, we have:
\[ b^2 = 48b \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ b^2 - 48b = 0 \]
Factoring out \( b \):
\[ b(b - 48) = 0 \]
This gives us two solutions: \( b = 0 \) or \( b = 48 \). Since \( b \) cannot be zero, we have:
\[ b = 48 \]

### Step 8: Find \( a \)
Now substitute \( b = 48 \) back into the equation for \( a \):
\[ a = \frac{3}{16} \times 48 \]
Calculating this gives:
\[ a = 9 \]

### Step 9: Find \( a + b \)
Now, we can find \( a + b \):
\[ a + b = 9 + 48 = 57 \]

### Final Answer
Thus, the value of \( a + b \) is:
\[ \boxed{57} \]