Answer for Day 6 Math Problem

9 seats 🎯

Since the unit which is "seats" is already included in the problem question, answers which are only "9" are also correct.

Solution

The formula for the average of interior angles of any polygon is the following:

a = (n - 2) * 180° / n

We will have 2 equations - the one before @jancharlest snuck a seat and the one after.

• Equation 1: a = (n - 2) * 180° / n
• Equation 2: a + 4 = (n - 2 + 1) * 180° / (n + 1)

where

• a = the average of the interior angles of the original polygon
• n = the number of sides of the original polygon

Simplifying Equation 2 to make the left side of the equation contain only a gives the following:

• Equation 3: a = (n - 1) * 180° / (n + 1) - 4

Since Equation 1 and Equation 3 have the same value on one side, we can now apply the Transitive Property of Equality, giving us the following:

• Equation 4: (n - 2) * 180° / n = (n - 1) * 180° / (n + 1) - 4

We shall now solve for the value of n in Equation 4.

• The problem is (n - 2) * 180 / n = (n - 1) * 180 / (n + 1) - 4.
• Multiplying both sides by the LCD which is n * (n + 1), we get (n - 2) * 180 * (n + 1) = (n - 1) * 180 * n - 4 * n * (n + 1).
• Distributing the terms, we get 180 * (n² - n - 2) = 180 * (n² - n) - 4 * (n² + n).
• Further distributing the terms, we get 180n² - 180n - 360 = 180n² - 180n - 4n² - 4n.
• Isolating zero on one side to form a quadratic equation, we get 4n² + 4n - 360 = 0. It can be further simplified into n² + n - 90 = 0.
• Solving for the value of n in the quadratic equation, we get n = 9 and n = -10, but there is no such as thing as negative number of sides, so we will stick with n = 9.

Winner: @jfang003 🏅

The reward of 1 HIVE has been sent to @jfang003's HIVE account. 💰

• I really got confused about @minus-pi's explanation. 🤯 Just for him, I typed the 3 paragraphs below! 😅

To understand the problem better, let's start with a quadrilateral, whose interior angles has an average of 90°. Adding 1 more side to the quadrilateral makes the average of its interior angles equal to 108°. In this situation, adding a side onto a quadrilateral makes the average of the polygon's interior angles increase by 18°.

To explain the above paragraph further: let's assume that the polygons are regular. The addition of a side to a square such that it becomes a regular pentagon will change each angle from 90° into 108°. These individual angle measures are the same as the averages of the interior angles of their respective regular polygons.

For me, @jfang003 understood the problem correctly. After a new vertex (a chair) has been added to the polygon (the set of chairs), the average of the interior angles increased by 4°. I absolutely could not understand the statement "the sum of interior angles increased [by some tiny amount such as 4]", since interior angles can be increased or decreased by denominations of 90°, such that their sum is at least 180°.

Mentions: @holovision, @ahmadmanga (@ahmadmangazap), @eturnerx (@eturnerx-dbuzz), @dkmathstats, @paultactico2, and @appukuttan66 🤓
Special mentions: @dbuzz, @chrisrice, @jancharlest, and @mehmetfix 🤯