If `(a)/(b)=(c)/(d)=(e)/(f)=3` , then `(2a^2 + 3c^(2) + 4e^(2))/(2b^(2) + 3d^(2) + 4f^(2))=?`

Options

A. 2
B. 3
C. 4
D. 9

Answer

To solve the problem, we start with the given ratios:

1. **Given Ratios**:
\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = 3 \]

This implies:
\[ a = 3b, \quad c = 3d, \quad e = 3f \]

2. **Substituting Values**:
We need to substitute these values into the expression:
\[ \frac{2a^2 + 3c^2 + 4e^2}{2b^2 + 3d^2 + 4f^2} \]

Substituting \(a\), \(c\), and \(e\):
\[ = \frac{2(3b)^2 + 3(3d)^2 + 4(3f)^2}{2b^2 + 3d^2 + 4f^2} \]

3. **Calculating the Numerator**:
Now, let's calculate the numerator:
\[ = \frac{2(9b^2) + 3(9d^2) + 4(9f^2)}{2b^2 + 3d^2 + 4f^2} \]
\[ = \frac{18b^2 + 27d^2 + 36f^2}{2b^2 + 3d^2 + 4f^2} \]

4. **Factoring Out the Common Term**:
Notice that we can factor out a 9 from the numerator:
\[ = \frac{9(2b^2 + 3d^2 + 4f^2)}{2b^2 + 3d^2 + 4f^2} \]

5. **Simplifying the Expression**:
Now, we can simplify the expression:
\[ = 9 \]

Thus, the final answer is:
\[ \boxed{9} \]