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What is the central angle (in degrees) corresponding to the total voters in village Z2 in the pie chart?

Question

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What is the central angle (in degrees) corresponding to the total voters in village Z2 in the pie chart?

Accepted Answer

**Solution:**

1. Let the central angles corresponding to total voters in villages Z2 and Z4 be $3x$° and $7x$° respectively, since their ratio is given as 3:7.

2. The total central angle for the pie chart is 360°. The problem states that the central angle for village Z3 is $(252° - 10x)$.

3. According to the condition, total voters in village Z3 are more than those in Z2, so:
   $$252° - 10x > 3x$$
   $$252° > 13x$$
   $$x < \frac{252}{13} \approx 19.38$$

4. We check the options for the central angle of village Z2 (which is $3x$°) to find a value of $x$ that satisfies the above inequality.

- For option (a) 72°: $3x = 72 \Rightarrow x = 24$ (Not valid as $x$ must be less than 19.38)
- For option (b) 54°: $3x = 54 \Rightarrow x = 18$ (Valid)
- For option (c) 60°: $3x = 60 \Rightarrow x = 20$ (Not valid)
- For option (d) 75°: $3x = 75 \Rightarrow x = 25$ (Not valid)

5. Only option (b) with $x=18$ satisfies the inequality.

6. Therefore, the central angles are:
   - Village Z2: $3x = 54°$
   - Village Z3: $252° - 10x = 252° - 180° = 72°$
   - Village Z4: $7x = 126°$

**Answer:** 54°

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