`(a)/(b)=(c )/(d)=(e )/(f)=(7)/(11)`
Find `(3a^(2)+5c^(2)-7c^(2))/(3b^(2)+5d^(2)-7f^(2))=?`

Answer

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

\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{7}{11} \]

### Step 1: Express variables in terms of a common ratio
Let’s denote the common ratio as \( k \). Then we can express \( a, b, c, d, e, \) and \( f \) as follows:

\[ a = 7k, \quad b = 11k \]
\[ c = 7k, \quad d = 11k \]
\[ e = 7k, \quad f = 11k \]

### Step 2: Substitute into the expression
Now, we substitute \( a, b, c, d, \) and \( f \) into the expression we need to evaluate:

\[ \frac{3a^2 + 5c^2 - 7c^2}{3b^2 + 5d^2 - 7f^2} \]

### Step 3: Simplify the numerator
First, simplify the numerator:

\[ 3a^2 + 5c^2 - 7c^2 = 3a^2 + (5c^2 - 7c^2) = 3a^2 - 2c^2 \]

Substituting \( a \) and \( c \):

\[ = 3(7k)^2 - 2(7k)^2 \]
\[ = 3 \cdot 49k^2 - 2 \cdot 49k^2 = (3 - 2) \cdot 49k^2 = 49k^2 \]

### Step 4: Simplify the denominator
Now, simplify the denominator:

\[ 3b^2 + 5d^2 - 7f^2 = 3b^2 + 5d^2 - 7f^2 \]

Substituting \( b, d, \) and \( f \):

\[ = 3(11k)^2 + 5(11k)^2 - 7(11k)^2 \]
\[ = (3 + 5 - 7) \cdot 121k^2 = 1 \cdot 121k^2 = 121k^2 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{49k^2}{121k^2} \]

### Step 6: Cancel \( k^2 \)
Since \( k^2 \) is common in both the numerator and denominator, we can cancel it out:

\[ = \frac{49}{121} \]

### Final Answer
Thus, the value of the expression is:

\[ \frac{49}{121} \]