Question

A library has received a single large contribution of \(\$ 5000 .\) A walkathon is also being held in which each sponsor will contribute \(\$ 10 .\) Let \(x\) represent the number of sponsors. Explain how such averages could be used to misrepresent actual contributions.

Short answer

The average can be used to misrepresent actual contributions by underrepresenting the contribution of each sponsor, because it dilutes each sponsor's contribution. Despite each sponsor only contributing $10, the average is inflated by the single large contribution of $5000.

Step-by-step solution

Understand the given quantities

We are given a large contribution of $5000 and walkathon sponsors who each give $10. Let's say the number of sponsors is \(x\). So, the total money contributed by the sponsors is \(10x\).

Calculate the total amount of contributions and the average contribution

Sum up the contributions to get the total amount contributed. This would be $5000 from the single large contribution plus the total contribution from the sponsors, which is \(10x\). The total amount is then $5000 + \(10x\). To find the average contribution, this total amount needs to be divided by the total number of contributors, which would be 1 (from the single large contribution) + \(x\) (the number of sponsors). The average contribution is then \(\frac{5000 + 10x}{1 + x}\).

Analyze how the average can be misleading

As the number of sponsors or \(x\) increases, the value of $10x$ increases, but because there's a large jumpstart with the $5000 contribution, the average doesn't completely reflect each sponsor's contribution. The more sponsors there are, the less each one effectively contributes to the average, and conversely the larger the single donation appears.

Key concepts

Algebraic Representation

Algebra is a powerful tool for representing and solving problems involving unknown quantities. In our exercise, algebra helps us model the situation where contributions to a library fund are being made. Let's consider the given scenario where the library has received a single contribution of \(5000 and an additional \)10 from each sponsor participating in a walkathon. We designate the variable \(x\) to represent the unknown number of sponsors.

Using algebra, we can express the total amount of money contributed through the walkathon as \(10x\), and when combined with the large initial contribution, the sum is \(5000 + 10x\). By introducing this algebraic representation, we create a mathematical model that reflects the real-world situation in a clear and quantifiable manner. It allows us to do further analysis on how the funds accumulate as more sponsors join, and also lays the foundation for understanding the average contribution per person, which we can see is influenced by both parameters: the fixed large donation and the variable sponsor contributions.

Average Calculation

Averages are an essential statistical tool, offering a way to summarize a set of numbers with a single value. To calculate the average contribution in our exercise, we need to divide the total amount of money contributed by the total number of contributors.

We've already seen that the total contribution is an algebraic sum \(5000 + 10x\) and when we divide this by the total number of contributors (the single donor plus the number of sponsors, \(1 + x\)), we have the formula for the average contribution per person: \[\frac{5000 + 10x}{1 + x}\]. While this formula provides a straightforward arithmetic mean, it doesn't effectively communicate the distribution of individual contributions, especially in cases where the number of contributors varies widely. As we'll see in the mathematical analysis, the average can be a misleading representation of the actual individual contributions.

Mathematical Analysis

Mathematical analysis allows us to understand deeper implications of the figures we calculate. Take the average contribution formula arrived at above: \[\frac{5000 + 10x}{1 + x}\]. It gives us an average that seems to describe how much each participant, on average, contributes to the fundraiser. However, this average can be misleading.

For example, if there is only one sponsor (\(x = 1\)), the average contribution would be \$2,505\, which is massively skewed by the large initial donation. As the number of sponsors increases, the effect of the \$5000\ becomes more diluted, lowering the average contribution—but the representation might still suggest that the contribution per person is more substantial than the \$10\ walkathon sponsors are actually giving. This could potentially be used to paint a deceptive picture of generosity in promotional materials or reports. Understanding the effect of the variables on the overall average is key in recognizing how averages can lead to misconceptions about individual contributions to the whole.