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

Options

A. 1
B. 4
C. 8
D. 16

Answer

To solve the problem step by step, we start with the given ratios and manipulate them to find the required expression.

### Step 1: Understand the Ratios
We are given that:
\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = 4 \]
From this, we can express \(a\), \(c\), and \(e\) in terms of \(b\), \(d\), and \(f\).

### Step 2: Express Variables
From the ratios, we can write:
1. \(a = 4b\)
2. \(c = 4d\)
3. \(e = 4f\)

### Step 3: Substitute into the Expression
We need to evaluate:
\[ \frac{m^2 a + n^2 c + p^2 e}{m^2 b + n^2 d + p^2 f} \]
Substituting the expressions for \(a\), \(c\), and \(e\):
\[ = \frac{m^2 (4b) + n^2 (4d) + p^2 (4f)}{m^2 b + n^2 d + p^2 f} \]

### Step 4: Factor Out the Common Term
In the numerator, we can factor out the 4:
\[ = \frac{4(m^2 b + n^2 d + p^2 f)}{m^2 b + n^2 d + p^2 f} \]

### Step 5: Simplify the Expression
Now, we can simplify the expression:
\[ = 4 \cdot \frac{(m^2 b + n^2 d + p^2 f)}{(m^2 b + n^2 d + p^2 f)} = 4 \]

### Final Answer
Thus, the value of the given expression is:
\[ \boxed{4} \]