Problem 71 - Prealgebra

Question

Find the mean and the range for each set of numbers. $$1,4,5,10,10$$

Step-by-Step Solution

Answer

The mean is 6, and the range is 9.

Step 1: Calculating the Mean - Step 1

To find the mean, first add up all the numbers in the set: \(1 + 4 + 5 + 10 + 10 = 30\).

Step 2: Calculating the Mean - Step 2

Next, divide the sum by the number of numbers in the set. There are 5 numbers: \(\frac{30}{5} = 6\). So, the mean is 6.

Step 3: Finding the Range - Step 3

Identify the smallest and largest numbers in the set. The smallest number is 1 and the largest is 10.

Step 4: Finding the Range - Step 4

Subtract the smallest number from the largest number to find the range: \(10 - 1 = 9\). So, the range is 9.

Key Concepts

Range Calculation

The concept of range is important when analyzing a set of numbers, as it helps you see the spread or difference between the highest and lowest values. To calculate the range, you need two key numbers: the smallest and the largest in your data set. For example, with the numbers provided in the original exercise, 1 is the smallest and 10 is the largest. These values are crucial because the range is simply the difference between them.

• Identify the smallest number (in this case, 1).
• Identify the largest number (in this case, 10).
• Subtract the smallest number from the largest: \(10 - 1 = 9\).

Thus, the range of this set is 9. This indicates how "spread out" the numbers are, providing insight into the variability of the data.

Step-by-Step Solution

Solving problems one step at a time can make computations more manageable and less intimidating, especially in prealgebra. Let's break down the calculation of the mean and the range as shown in the solution.
First, to find the mean:

• Add all numbers in the set: \(1 + 4 + 5 + 10 + 10 = 30\).
• Count the numbers in the set (there are 5 numbers).
• Divide the sum by the number of numbers: \(\frac{30}{5} = 6\).

This step-by-step breakdown makes it easy to follow each calculation, ensuring nothing is overlooked. Now, to find the range, the steps are simple too:

• Identify smallest (1) and largest (10) numbers.
• Subtract the smallest from the largest: \(10 - 1 = 9\).

By handling each step separately, you minimize mistakes and build confidence in your arithmetic skills.

Prealgebra Concepts

Prealgebra is the foundation of mathematics that prepares students for more advanced algebra. It involves understanding key numerical concepts, such as mean and range, which are simple calculations that provide valuable insights into a set of data.

• The mean, or average, gives an idea of the "central" value among numbers. It's computed by adding all numbers together and dividing by how many numbers there are.
• The range shows how spread out the numbers are, indicating variance by showing the distance between the smallest and largest values.

These basic concepts are crucial and often the starting point for more complex studies in mathematics. They make comparisons between different sets of numbers easier and clearer, helping to develop analytical skills that are applicable throughout one's education.