Problem 11 - College Algebra with Modeling and Visualization

Question

Find a set of three numbers with a mean of 20 and a median of \(18 .\) Is your answer unique?

Step-by-Step Solution

Answer

The answer is not unique; multiple sets satisfy the conditions.

Step 1: Understanding the Mean

The mean of three numbers is the sum of the numbers divided by three. If the mean is 20, then the sum of the three numbers must be \(20 \times 3 = 60\). Let these numbers be \(a\), \(b\), and \(c\) where \(a \leq b \leq c\). So, \(a + b + c = 60\).

Step 2: Understanding the Median

The median of three numbers \(a\), \(b\), and \(c\), arranged in ascending order, is the middle number \(b\). It is given that the median is 18, so \(b = 18\).

Step 3: Forming Equations

From the conditions we have, \(a + 18 + c = 60\). Simplifying gives \(a + c = 42\). Now we need to find integers \(a\) and \(c\) such that they satisfy all conditions and the inequalities \(a \leq 18 \leq c\).

Step 4: Finding Solutions

Since the sum of \(a\) and \(c\) is 42, we can find pairs of \(a\) and \(c\). We need to ensure that \(a \leq 18 \leq c\). For example, if \(a = 15\), then \(c = 27\); if \(a = 16\), then \(c = 26\), and if \(a = 18\), then \(c = 24\). These satisfy the condition \(a \leq 18 \leq c\).

Step 5: Checking for Uniqueness

The solutions include pairs: \((15, 18, 27)\), \((16, 18, 26)\), and \((18, 18, 24)\). Since there are multiple sets of numbers satisfying the given conditions, the answer is not unique.

Key Concepts

Understanding Mean

The mean, often referred to as the average, is a fundamental concept in algebra that helps us understand data sets. It is calculated by taking the sum of all numbers in a set and dividing it by the total number of numbers. This provides a central value that represents the data set as a whole.
For a set of three numbers, the formula for the mean can be expressed as:

• Mean = \( \frac{a + b + c}{3} \)

In our exercise, the mean is given as 20. Therefore, the equation becomes \( \frac{a + b + c}{3} = 20 \). By solving this, you find that \(a + b + c = 60\).
Understanding this helps you see how the mean affects the sum of numbers in a set and gives insights into how changes in individual numbers will influence the mean.

Grasping the Median

The median provides a measure of central tendency like the mean, but it focuses on the middle value in an ordered data set. In our context, when dealing with three numbers \(a\), \(b\), and \(c\) arranged in increasing order, the median is the middle number: \(b\).
The advantage of the median over the mean is that it is less affected by outliers or extremely large or small values. To find the median in this exercise, we set the middle value \(b\) to 18, as given. Thus, from \(a \leq b \leq c\), \(b\) becomes 18.
This shows how the median defines the placement of other numbers around it, meaning \(c\) must be equal to or larger than 18 and \(a\) must be equal to or smaller than 18. The median offers a simple, yet powerful way of understanding the distribution of a set.

Exploring Inequality

Inequality in algebra refers to mathematical expressions that denote a relationship of non-equality between two values. It is crucial, especially when arranging or comparing numbers. In this exercise, we see inequality in action through the conditions:

• \(a \leq b \leq c\)
• \(a + b + c = 60\)

These inequalities provide constraints that must be met while solving for possible sets of numbers.
By applying these, we ensure that our set of numbers meets the criteria for both mean and median while still respecting the order imposed by the inequality. For instance, adjusting \(a\) and \(c\) to satisfy \(a + c = 42\) while maintaining \(a \leq 18\) and \(c \geq 18\) gives us several solutions, such as \((15, 18, 27)\), \((16, 18, 26)\), and \((18, 18, 24)\).
This exercise demonstrates the use of inequalities to find solutions under specific conditions, offering a flexible approach to problem-solving in algebra.