Solving for the Third Number Given Specific Averages

Understanding the concept of the average (arithmetic mean) is a fundamental tool in solving mathematical problems. In the following example, we will determine the third number in a set of five numbers, given specific conditions about their averages.

Problem Statement

Given that the average of five numbers is 54, the average of the first and second numbers is 46, and the average of the fourth and fifth numbers is 33, we need to find the third number.

Step-by-Step Solution

1. Determine the Sum of the Five Numbers

The average of five numbers is 54. Therefore, the sum of these five numbers can be calculated as follows:

Average Sum of Numbers / Number of Numbers

54 Sum of Five Numbers / 5

Sum of Five Numbers 54 x 5 172

2. Calculate the Sum of the First and Second Numbers

Given that the average of the first and second numbers is 46, we can determine the sum of these two numbers as:

Average Sum of Numbers / Number of Numbers

46 Sum of First and Second Numbers / 2

Sum of First and Second Numbers 46 x 2 93

3. Calculate the Sum of the Fourth and Fifth Numbers

Given that the average of the fourth and fifth numbers is 33, we can determine the sum of these two numbers as:

Average Sum of Numbers / Number of Numbers

33 Sum of Fourth and Fifth Numbers / 2

Sum of Fourth and Fifth Numbers 33 x 2 66

4. Calculate the Sum of the First, Second, Fourth, and Fifth Numbers

Adding the sums of the first, second, and fourth and fifth numbers:

93 66 159

5. Determine the Third Number

Subtract the sum of the first, second, fourth, and fifth numbers from the total sum of the five numbers to find the third number:

172 - 159 43

Therefore, the third number is 43.

Verification Through Algebraic Method

Let's denote the five numbers as a, b, c, d, e.

1. The sum of the five numbers is:

a b c d e 34.4 * 5 172 —— (1)

2. The sum of the first and second numbers is:

a b 46.5 * 2 93 —— (2)

3. The sum of the fourth and fifth numbers is:

d e 18 * 2 36 —— (3)

Adding equations (2) and (3) and subtracting from (1):

93 c 36 172

129 c 172

c 172 - 129 43

Conclusion

The third number, denoted as c, is 43.

Key Concepts

Average (or arithmetic mean) is the sum of a set of numbers divided by the count of numbers. This understanding helps in solving a wide range of mathematical problems involving averages.

Through consistent practice, you can better manage and solve similar problems in the future. If you found this explanation helpful, please consider upvoting and sharing to help others too!