To solve for $a$ in this pie chart with the given ratio $a : b : c = 4 : 3 : 5$, let's follow these steps:

1. The angles in a pie chart must add up to $360^\circ$ because it represents a complete circle.

2. Given $a : b : c = 4 : 3 : 5$, we can let: $a = 4x, \quad b = 3x, \quad c = 5x$ where $x$ is a common multiplier.

3. Since the sum of the angles $a + b + c = 360^\circ$, we have: $4x + 3x + 5x = 360$ Simplifying this, we get: $12x = 360$ Solving for $x$, we divide both sides by 12: $x = 30$

4. Now we can find $a$ by substituting $x = 30$ into $a = 4x$: $a = 4 \times 30 = 120^\circ$

Thus, the value of $a$ is $120^\circ$.

Would you like further details or have any questions about this calculation?

Here are some related questions to explore further:

1. How would you calculate $b$ and $c$ given the same ratio?
2. If the total angle were different, say $400^\circ$, how would that affect the value of $a$?
3. What percentage of the total survey does each opinion category represent?
4. How can you verify that the ratio $4 : 3 : 5$ is accurate once you find $a$, $b$, and $c$?
5. What are other practical applications of ratios in interpreting survey data?

Tip: Always check if the sum of the angles matches the expected total in a pie chart to ensure accurate calculations.