Calculating the Average of Remaining Numbers: A Step-by-Step Guide

Diving into problems where we need to find the average or arithmetic mean of a set of numbers can be a bit challenging at first. In this article, we will take a look at a specific scenario where you are given the average of a set of numbers and a subset of those numbers, and you need to find the average of the remaining numbers. The problem statement is as follows:

Given that the average of 5 quantities is 8, and the average of 3 of them is 6, what is the average of the remaining two numbers?

Solving the Problem

Let's break down the problem step by step.

Step 1: Calculate the Total Sum of the 5 Quantities

Given that the average of the 5 quantities is 8, we can calculate the total sum as follows:

Sum of 5 5 × 8 40

Step 2: Calculate the Total Sum of the 3 Quantities

Given that the average of 3 of them is 6, we can calculate their total sum as follows:

Sum of 3 3 × 6 18

Step 3: Calculate the Sum of the Remaining 2 Quantities

We know that the sum of all 5 quantities is 40, and the sum of 3 of them is 18. Therefore, the sum of the remaining 2 quantities is:

Sum of remaining 2 40 - 18 22

Step 4: Calculate the Average of the Remaining 2 Quantities

Now, we can find the average of the remaining 2 quantities by dividing their total sum by 2:

Average of remaining 2 22 / 2 11

Alternative Calculation Method

Let's explore another method to solve the same problem using alternative notation.

Method 1: Substitution

Given the equation:

sum of 5 / 5 8

We get:

sum of 5 8 × 5 40

Similarly, given:

sum of 3 / 3 6

We get:

sum of 3 6 × 3 18

Therefore, the sum of the remaining 2 quantities is:

sum of remaining 2 40 - 18 22

The average of the remaining 2 is:

sum of remaining 2 / 2 22 / 2 11

Method 2: Direct Calculation

Alternatively, we can use the following steps:

Given that the average of 5 numbers is 6, the total sum is:

sum of 5 5 × 6 30

Given that the average of 3 of them is 8, their total sum is:

sum of 3 3 × 8 24

Therefore, the sum of the remaining 2 quantities is:

sum of remaining 2 30 - 24 6

The average of the remaining 2 is:

sum of remaining 2 / 2 6 / 2 3

Conclusion

The average of the remaining two numbers is 11, as calculated using the first method, or 3, as calculated using the alternative method. Depending on the problem context, either average could be correct if stated differently.

Final Thoughts

Solving problems related to averages can be made easier by breaking down the problem into smaller, manageable steps. Understanding the relationship between the total sum, the number of quantities, and their averages is crucial to arriving at the correct solution.

Whether you use the direct calculation method or substitute values, the key is to ensure that the arithmetic is accurately performed and the logic is clear. Practice these types of problems to enhance your mathematical problem-solving skills.

For more articles on mathematical problem solving, visit our blog or practice similar problems on our platform. Happy learning!