if `(a)/(b)=(c)/(d)=(e)/(f)=(1)/(3)`, then `(5a+3c+7e)/(5b+3d+7f)=`_________

Answer

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

\[ \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{1}{3} \]

From this, we can express \(a\), \(c\), and \(e\) in terms of \(b\), \(d\), and \(f\) respectively.

1. **Express \(a\), \(c\), and \(e\)**:
- From \(\frac{a}{b} = \frac{1}{3}\), we can write:
\[ a = \frac{1}{3}b \]
- From \(\frac{c}{d} = \frac{1}{3}\), we can write:
\[ c = \frac{1}{3}d \]
- From \(\frac{e}{f} = \frac{1}{3}\), we can write:
\[ e = \frac{1}{3}f \]

2. **Substituting \(a\), \(c\), and \(e\) into the expression**:
We need to find the value of:
\[ \frac{5a + 3c + 7e}{5b + 3d + 7f} \]
Substituting the expressions we found:
\[ 5a = 5 \left(\frac{1}{3}b\right) = \frac{5}{3}b \]
\[ 3c = 3 \left(\frac{1}{3}d\right) = d \]
\[ 7e = 7 \left(\frac{1}{3}f\right) = \frac{7}{3}f \]

Now substituting these back into the numerator:
\[ 5a + 3c + 7e = \frac{5}{3}b + d + \frac{7}{3}f \]

3. **Finding a common denominator for the numerator**:
The common denominator for the terms in the numerator is 3:
\[ \frac{5}{3}b + d + \frac{7}{3}f = \frac{5b + 3d + 7f}{3} \]

4. **Now, substitute into the denominator**:
The denominator is:
\[ 5b + 3d + 7f \]

5. **Putting it all together**:
Now we can write the entire expression:
\[ \frac{5a + 3c + 7e}{5b + 3d + 7f} = \frac{\frac{5b + 3d + 7f}{3}}{5b + 3d + 7f} \]

This simplifies to:
\[ \frac{5b + 3d + 7f}{3(5b + 3d + 7f)} = \frac{1}{3} \]

Thus, the final answer is:
\[ \frac{5a + 3c + 7e}{5b + 3d + 7f} = \frac{1}{3} \]