Finding the Fourth Number: A Comprehensive Guide to Solving Averaging Problems

When dealing with problem-solving in mathematics, especially with averages, it can sometimes feel like a daunting task. However, with a few simple steps, you can easily find the missing number. In this article, we'll guide you through a step-by-step process to find the fourth number in a series where the overall and segmented averages are given. We'll explore various solutions and provide a deep understanding of the underlying concepts.

The Basics of Averages and Summations

In the context of averages, the fundamental formula is crucial:

Average sum of all terms / number of terms

Solution Approach 1

Let's begin with the first solution to a typical problem involving the average of 6 numbers. Initially, the problem states that the average of 6 numbers is 30, and the average of the first four numbers is 25. Additionally, the average of the last three numbers is 35. Our goal is to find the fourth number.

Calculate the total sum of all 6 numbers: Total of 6 numbers 6 × 30 180 Calculate the sum of the first four numbers: Total of the first four numbers 25 × 4 100 Calculate the sum of the last three numbers: Total of the last three numbers 35 × 3 105 Find the sum of the first five numbers: Total of the first five numbers 100 (the fourth number to find) Calculate the sixth number: Sum of all 6 numbers Sum of the first 5 numbers the fourth number 180 125 105 - 100 the fourth number 180 125 80 the fourth number 40 the fourth number

Solution Approach 2

Another method involves a more straightforward subtraction approach:

Use the overall average: Total of 6 numbers 6 × 30 180 Find the sum of the first four numbers: Total of the first four numbers 4 × 25 100 Find the sum of the last three numbers: Total of the last three numbers 3 × 35 105 Calculate the sum of the first four and last three numbers: Sum of the first four and last three numbers 100 105 - (the fourth number) 100 105 - the fourth number 180 - 100 105 - the fourth number 80 The fourth number 105 - 80 25

A Solving Technique with Variables

For a more algebraic approach, let the fourth number be represented as x:

Set up the equation based on the overall total: Total of the six numbers 30 × 6 180 Express the total of the first five numbers: Total of the first five numbers 123 45 123 x 100 (as the sum of the first four is 100) Express the total of the last three numbers: Total of the last three numbers 456 105 (as it is given that the average of the last three is 35) Solve for x: 180 123 x 105 - 100 - 105 100 x 180 210 - 180 30 x 210 - 180 30

Understanding the Process

The key to solving these types of problems is breaking down the problem into manageable parts and using the properties of averages and sums. By understanding the formulas and how they interact, you can easily find the missing number in a series of averages.

Conclusion

Solving problems involving averages can be straightforward if you follow a structured approach. Whether you use direct calculations, subtraction, or algebraic methods, the goal is to find the missing piece of information. Understanding the basic principles of averages and applying them systematically will help you solve similar problems in the future.

Further Reading

We recommend exploring more problems like this to solidify your understanding. You can find similar problems and solutions on educational websites and mathematics forums. Keeping a notebook of these types of problems will also help you practice and improve your skills.