Solving Pie Chart Central Angles Problem

This problem involves finding the values of central angles in a pie chart based on a given relationship and then calculating a specific expression involving those angles. A pie chart represents a whole, and the sum of the central angles of all sectors in a pie chart is always 360 degrees.

Understanding the Problem Statement

We are given four central angles of sectors in a pie chart, denoted as p, q, r, and s, in degrees. The total measure of these angles must sum up to 360 degrees. We are also given a specific relationship between these angles: \(9p = 3q = 2r = 6s\).

Setting up the Equations

From the problem description, we have two key pieces of information:

1. The sum of the central angles in a pie chart is 360 degrees: \(p + q + r + s = 360\).
2. The given relationship between the angles: \(9p = 3q = 2r = 6s\).

Solving for the Angles

To find the values of p, q, r, and s, we can use the given relationship. Let's set the common value of the relationship to a constant, say \(k\).

\(9p = k \implies p = \frac{k}{9}\)

\(3q = k \implies q = \frac{k}{3}\)

\(2r = k \implies r = \frac{k}{2}\)

\(6s = k \implies s = \frac{k}{6}\)

Now, substitute these expressions in terms of \(k\) into the sum of angles equation:

\(\frac{k}{9} + \frac{k}{3} + \frac{k}{2} + \frac{k}{6} = 360\)

To solve for \(k\), we find a common denominator for the fractions, which is 18.

\(\frac{2k}{18} + \frac{6k}{18} + \frac{9k}{18} + \frac{3k}{18} = 360\)

\(\frac{2k + 6k + 9k + 3k}{18} = 360\)

\(\frac{20k}{18} = 360\)

\(\frac{10k}{9} = 360\)

\(10k = 360 \times 9\)

\(10k = 3240\)

\(k = \frac{3240}{10}\)

\(k = 324\)

Now that we have the value of \(k\), we can find the individual central angles:

\(p = \frac{k}{9} = \frac{324}{9} = 36\)

\(q = \frac{k}{3} = \frac{324}{3} = 108\)

\(r = \frac{k}{2} = \frac{324}{2} = 162\)

\(s = \frac{k}{6} = \frac{324}{6} = 54\)

Let's verify that the sum of these angles is 360 degrees:

\(36 + 108 + 162 + 54 = 360\). The angles are correct.

Calculating the Expression \(4p - q\)

The question asks for the value of the expression \(4p - q\). We have found that \(p = 36\) and \(q = 108\).

\(4p - q = 4(36) - 108\)

\(4 \times 36 = 144\)

So, the expression becomes:

\(144 - 108 = 36\)

The value of \(4p - q\) is 36.

Summary of Steps

1. Identify that the sum of central angles in a pie chart is 360 degrees.
2. Use the given relationship \(9p = 3q = 2r = 6s\) to express each angle in terms of a common constant \(k\).
3. Substitute these expressions into the sum equation \(p + q + r + s = 360\) and solve for \(k\).
4. Calculate the values of p and q using the found value of \(k\).
5. Calculate the value of the expression \(4p - q\).

Revision Table: Pie Chart Angles

Additional Information: Central Angles and Proportions

In a pie chart, the central angle of a sector is proportional to the quantity it represents. If a sector represents a fraction \(f\) of the total, its central angle is \(f \times 360^\circ\). The given relationship \(9p = 3q = 2r = 6s\) implies that the quantities represented by sectors p, q, r, and s are in specific proportions. For example, since \(9p = 3q\), \(p/q = 3/9 = 1/3\), meaning angle \(q\) is 3 times angle \(p\). Similarly, \(3q = 2r \implies q/r = 2/3\), and so on. These proportional relationships are key to understanding how the angles are distributed within the total 360 degrees.