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Chapter 2: Problem 44

A pie chart shows the market shares of cola products. The "slice" for Pepsi- Cola has a central angle of 90 degrees. What is their market share?

Short Answer

Expert verified
Pepsi-Cola's market share is 25%.

Step by step solution

01

Understanding the Problem

In this problem, we need to determine the fraction of the whole market share that Pepsi-Cola represents if its slice in the pie chart has a central angle of 90 degrees.
02

Total Degrees in a Circle

Recall that a pie chart represents a full circle, which is always 360 degrees. This means the entire market share is represented by a total angle of 360 degrees.
03

Calculate the Fraction

To find Pepsi-Cola's market share in fraction form, divide its central angle by the total degrees in the circle. Thus, the fraction is given by: \( \frac{90}{360} \).
04

Simplify the Fraction

Simplify \( \frac{90}{360} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 90. This gives: \( \frac{90}{360} = \frac{1}{4} \).
05

Convert Fraction to Percentage

To express Pepsi-Cola's market share as a percentage, multiply the simplified fraction by 100: \( \frac{1}{4} \times 100 = 25\% \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Market Share Calculation
Understanding the concept of market share in the context of a pie chart can be interesting and straightforward. In a pie chart, different segments or 'slices' represent portions of a whole. Each slice corresponds to a central angle. This is a reflection of a specific share of the market. To calculate market share, we compare this angle to the entire circle. The key is knowing that a full circle is always 360 degrees. For example, if Pepsi-Cola's slice is 90 degrees, that angle helps us determine its market share within the overall market displayed by the pie chart. In simple steps:
  • Recognize that the whole market is represented by 360 degrees.
  • Take the central angle of the specific slice鈥攊n this case, Pepsi-Cola's 90 degrees.
  • The market share as a fraction is calculated by dividing Pepsi's angle by 360.
The resulting fraction is a mathematical representation of Pepsi-Cola's share of the market.
Fraction Simplification
Once you have the fraction of the market share, simplifying it is crucial.This involves reducing it to the simplest form, which means making the numbers smaller but still representing the same value.To do this, find the greatest common divisor (GCD) of both the numerator and the denominator. For our market share calculation, we determined the fraction as \( \frac{90}{360} \).To simplify this fraction, we need to identify the GCD of 90 and 360, which is 90 itself.By dividing both the numerator and denominator by this GCD, we simplify:
  • The numerator becomes \( \frac{90}{90} = 1 \).
  • The denominator becomes \( \frac{360}{90} = 4 \).
Hence, the simplified fraction becomes \( \frac{1}{4} \), representing the same market share but in a more concise, easier-to-read form.
Percentage Conversion
Converting a fraction to a percentage is a common mathematical task, especially useful in visualizing parts of a whole.The key is multiplying the fraction by 100, which scales the fraction up to the percentage format.For Pepsi-Cola鈥檚 market share:Given the simplified fraction of \( \frac{1}{4} \), the conversion process involves:
  • Multiply the fraction \( \frac{1}{4} \) by 100.
  • This results in \( 25\%, \) meaning \( \frac{1}{4} \) of the circle (or market) is equal to 25%.
This conversion gives a clear, intuitive understanding of market share, making it easier to compare or analyze within the pie chart context.This step transforms a mathematical value into a format that is simple and understandable for everyday use.

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Most popular questions from this chapter

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