If $a : b = c : d = e : f = 5 : 7$, then what is the ratio $\left(3a + 5c + 11e\right) : \left(3b + 5d + 11f\right)$?

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**Solution:**

1. Given that:
   $$a : b = c : d = e : f = 5 : 7$$

This means:
   $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{5}{7}$$

2. From the ratio, express each pair in terms of a common variable $k$:
   - Let $a = 5k$, $b = 7k$
   - Let $c = 5m$, $d = 7m$
   - Let $e = 5n$, $f = 7n$

3. Substitute these into the expression:
   $$3a + 5c + 11e = 3(5k) + 5(5m) + 11(5n) = 15k + 25m + 55n$$

$$3b + 5d + 11f = 3(7k) + 5(7m) + 11(7n) = 21k + 35m + 77n$$

4. Factor out 5 from numerator and 7 from denominator:
   $$\frac{15k + 25m + 55n}{21k + 35m + 77n} = \frac{5(3k + 5m + 11n)}{7(3k + 5m + 11n)}$$

5. Since $(3k + 5m + 11n)$ is common and non-zero, it cancels out:
   $$= \frac{5}{7}$$

**Answer:** $5 : 7$

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