Introduction

In the realm of basic arithmetic, determining the third number when given an average can be a straightforward and insightful exercise. This article will delve into the methodology and provide solutions for given problems. Let's explore how to find the third number in a series when the averages of the numbers and the first two are known.

Understanding the Concept of Average

The average (arithmetic mean) of a set of numbers is calculated by summing the numbers and then dividing by the count of numbers. The formula for the average of three numbers is given as: [ m frac{a b c}{3} ] Where (a), (b), and (c) are the three numbers, and (m) is the average. If we rearrange this formula, we can express (c) as follows: [ c 3m - a - b ]

Problem Solving Methods

Given the average (17) of three numbers, and the average (16) of the first two numbers, we can determine the third number using the following steps:

Start with the given formula for the average of three numbers: [ frac{a b c}{3} 17 ] Multiply both sides by 3 to isolate the sum of the numbers: [ a b c 51 ] Given that the average of the first two numbers is 16, we can express this as: [ frac{a b}{2} 16 ] Multiply both sides by 2 to express the sum of the first two numbers as: [ a b 32 ] Substitute the sum of the first two numbers into the equation for the sum of all three numbers: [ 32 c 51 ] Solving for (c): [ c 51 - 32 19 ]

This method illustrates a clear, step-by-step approach to finding the third number when the averages are provided.

General Formula

The problem can be generalized to any set of numbers where the average is known, and two out of the three numbers are given. The third number can be found using the formula:

[ c 3m - a - b ]

Where (m) is the average of the three numbers, and (a) and (b) are the known numbers.

Additional Proofs and Examples

Let's consider another example to solidify the understanding:

Given the sum of three numbers 16, 17, and 18 is 51, and the average is 17. To find the third number when two of the numbers (16 and 17) are known:

Calculate the sum of the known numbers: [ 16 17 33 ] Subtract this sum from the total sum of the numbers: [ 51 - 33 18 ] The third number is therefore: [ 19 ]

This example further reinforces the formula and the step-by-step approach used to find the third number.

Conclusion

By applying the method of averaging and substitution, finding the third number when the average and two numbers are given becomes a straightforward process. Whether dealing with simple arithmetic problems or more complex scenarios, the formula and logical steps remain consistent and reliable.

Key takeaways include the formula ( c 3m - a - b ) for finding the third number and the importance of understanding the basic properties of averages in arithmetic.