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Number sets are collections of numbers grouped together based on a specific rule or characteristic. They can be finite or infinite, and they help in understanding mathematical relationships and operations. In this exercise, we are working with a finite set of five numbers. A number set can include

• natural numbers,
• integers,
• rational numbers,
• and real numbers.

In our situation, we aim to form a set of five numbers whose arithmetic mean is 50. This requires the total sum of all numbers to be 250. While choosing numbers for our set, ensure they meet this criterion.
Understanding number sets allows flexibility, giving you options like using negative numbers, fractions, or decimal numbers. However, for simplicity, whole numbers are used in this exercise.

The sum of numbers is the result of adding numbers together. When calculating an arithmetic mean, understanding the sum is critical. In this exercise, the sum of the five numbers must be 250 to achieve a mean of 50.
To calculate the sum:

• Choose your numbers. In this example: 40, 50, 60, 70.
• Calculate their sum: \(40 + 50 + 60 + 70 = 220\).
• Determine how much to add to reach 250. The missing number would be \(250 - 220 = 30\).

The arithmetic mean involves dividing this sum by the total number of values in the set. Here, \(250 \div 5 = 50\), confirming the mean is indeed 50. This process reflects the importance of accurate addition in determining sums and means.

Basic algebra involves manipulating numbers and symbols to solve equations or simplify expressions. It's the backbone of many mathematical concepts and calculations. In this exercise, algebra helps us determine the unknown number needed to reach a specific sum.
Here's how algebra is applied:

• First, define the total sum required: \(5 \times 50 = 250\) for the mean goal.
• Then, subtract the sum of chosen numbers: \(250 - 220 = 30\). This gives the fifth number needed.

This use of algebra allows you to isolate an unknown variable—in this case, the fifth number—simplifying problem-solving. Recognizing how algebra connects different arithmetic operations makes it a powerful tool in math.