Calculating the Average of the Remaining Numbers

This problem requires us to use the concept of averages and sums to find the average of a subset of numbers when the total average and the averages of other subsets are known. The average of a set of numbers is defined as the sum of the numbers divided by the count of the numbers.

The formula for the average is:

\(\text{Average} = \frac{\text{Sum of numbers}}{\text{Number of terms}}\)

From this, we can find the sum of numbers if we know the average and the number of terms:

\(\text{Sum of numbers} = \text{Average} \times \text{Number of terms}\)

Step-by-Step Calculation for the Average Problem

We are given the following information:

• Total number of terms: 6
• Average of the six numbers: 3.52
• Average of the first two numbers: 3.7
• Average of the next two numbers: 2.5
• We need to find the average of the remaining two numbers.

Let's calculate the sums based on the given averages:

1. Calculate the total sum of the six numbers:

Sum of 6 numbers = Average of 6 numbers \(\times\) Number of terms

Sum of 6 numbers = \(3.52 \times 6\)

Sum of 6 numbers = \(21.12\)

2. Calculate the sum of the first two numbers:

Sum of first 2 numbers = Average of first 2 numbers \(\times\) Number of terms

Sum of first 2 numbers = \(3.7 \times 2\)

Sum of first 2 numbers = \(7.4\)

3. Calculate the sum of the next two numbers:

Sum of next 2 numbers = Average of next 2 numbers \(\times\) Number of terms

Sum of next 2 numbers = \(2.5 \times 2\)

Sum of next 2 numbers = \(5.0\)

4. Calculate the sum of the remaining two numbers:

The remaining numbers are the total numbers minus the first two and the next two. The number of remaining terms is \(6 - 2 - 2 = 2\). The sum of these remaining two numbers can be found by subtracting the sums of the known subsets from the total sum.

Sum of remaining 2 numbers = (Sum of 6 numbers) - (Sum of first 2 numbers) - (Sum of next 2 numbers)

Sum of remaining 2 numbers = \(21.12 - 7.4 - 5.0\)

Sum of remaining 2 numbers = \(21.12 - (7.4 + 5.0)\)

Sum of remaining 2 numbers = \(21.12 - 12.4\)

Sum of remaining 2 numbers = \(8.72\)

5. Calculate the average of the remaining two numbers:

Average of remaining 2 numbers = \(\frac{\text{Sum of remaining 2 numbers}}{\text{Number of remaining terms}}\)

Average of remaining 2 numbers = \(\frac{8.72}{2}\)

Average of remaining 2 numbers = \(4.36\)

So, the average of the remaining two numbers is 4.36.

Summary of Calculations

Average of remaining two numbers = \(\frac{8.72}{2} = 4.36\)

Revision Table: Key Average Concepts

Additional Information: Understanding Averages

The average, or arithmetic mean, is a fundamental concept in statistics and data analysis. It gives us a single value that summarizes a group of values. When we talk about the average of six numbers being 3.52, it means that if all six numbers were equal, they would each be 3.52. The sum of these six numbers is the same as the sum of six numbers each equal to 3.52.

Problems like this one, where you are given averages of different parts of a dataset and asked to find the average of the remaining part, are common in tests. The key is always to convert averages into sums, manipulate the sums, and then convert the resulting sum back into an average.

It's important to remember that the average of a group of numbers depends on both the sum of the numbers and how many numbers are in the group. Removing numbers with a lower average from a group will increase the average of the remaining numbers, while removing numbers with a higher average will decrease the average of the remaining numbers. In this case, the first two numbers have an average (3.7) higher than the total average (3.52), and the next two numbers have an average (2.5) lower than the total average. However, their combined sum needs to be considered relative to the total sum to find the remaining average.