Ratio and Proportion Setup

We are given the relationship between three quantities a, b, and c as:

$ \text{a} : 3 = \text{b} : 7 = \text{c} : 9 $

This can be rewritten in fractional form:

$ \frac{\text{a}}{3} = \frac{\text{b}}{7} = \frac{\text{c}}{9} $

Let's introduce a constant of proportionality, $k$.

$ \frac{\text{a}}{3} = k \quad \Rightarrow \quad \text{a} = 3k $

$ \frac{\text{b}}{7} = k \quad \Rightarrow \quad \text{b} = 7k $

$ \frac{\text{c}}{9} = k \quad \Rightarrow \quad \text{c} = 9k $

Expression Evaluation

We need to find the value of the expression $\frac{\text{a}+\text{b}+\text{c}}{\text{a}}$.

First, find the sum $\text{a}+\text{b}+\text{c}$ in terms of $k$:

$ \text{a}+\text{b}+\text{c} = 3k + 7k + 9k = 19k $

Now substitute the values of $\text{a}+\text{b}+\text{c}$ and $\text{a}$ into the expression:

$ \frac{\text{a}+\text{b}+\text{c}}{\text{a}} = \frac{19k}{3k} $

Assuming $k \neq 0$, we can cancel $k$ from the numerator and the denominator:

$ \frac{19k}{3k} = \frac{19}{3} $

Therefore, the value of the expression is $\frac{19}{3}$.