Study the pie chart and answer the question that follows.
What is the approximate difference between the area (in degrees) of Pool A and Pool B?
The question asks for the approximate difference in degrees between two pools of companies based on their market share. Pool A consists of the 3 companies with the largest market share, and Pool B includes the remaining companies.
Market Share Degree Calculation
A full circle represents a total market share of 100% or $360^\circ$. Therefore, 1% of the market share corresponds to $3.6^\circ$ ($360^\circ / 100$).
Pool Definitions
- Pool A: The 3 companies with the largest market share.
- Pool B: The remaining companies.
Calculation Approach
To find the difference in degrees:
- Determine the total market share percentage ($ P_A $) for the 3 companies in Pool A. This requires the specific market share data which is not provided.
- Convert Pool A's market share percentage to degrees: $ \text{Degrees}_A = P_A \times 3.6 $.
- Calculate the market share percentage for Pool B: $ P_B = 100\% - P_A $.
- Convert Pool B's market share percentage to degrees: $ \text{Degrees}_B = P_B \times 3.6 $ (or $ \text{Degrees}_B = 360^\circ - \text{Degrees}_A $).
- Calculate the absolute difference: $ | \text{Degrees}_A - \text{Degrees}_B | $.
Applying the Provided Answer
Since the specific market share data is missing, we can work backward from the given correct answer ($213^\circ$) to understand the relationship.
Let $ \text{Degrees}_A $ be the total degrees for Pool A and $ \text{Degrees}_B $ be the total degrees for Pool B.
We have two equations:
- $ \text{Degrees}_A + \text{Degrees}_B = 360^\circ $ (Total degrees in a circle)
- $ | \text{Degrees}_A - \text{Degrees}_B | = 213^\circ $ (The required difference)
Assuming Pool A has a larger share than Pool B ($ \text{Degrees}_A > \text{Degrees}_B $):
$ \text{Degrees}_A - \text{Degrees}_B = 213^\circ $
Add the two equations:
$ (\text{Degrees}_A + \text{Degrees}_B) + (\text{Degrees}_A - \text{Degrees}_B) = 360^\circ + 213^\circ $
$ 2 \times \text{Degrees}_A = 573^\circ $
$ \text{Degrees}_A = \frac{573^\circ}{2} = 286.5^\circ $
Now, find $ \text{Degrees}_B $:
$ \text{Degrees}_B = 360^\circ - \text{Degrees}_A = 360^\circ - 286.5^\circ = 73.5^\circ $
The difference is $ 286.5^\circ - 73.5^\circ = 213^\circ $, which matches the provided answer.
This calculation shows that Pool A represents approximately $ \frac{286.5}{3.6} \approx 79.6\% $ of the market share, and Pool B represents approximately $ \frac{73.5}{3.6} \approx 20.4\% $. The difference of $213^\circ$ is consistent with these proportions.
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