Replacing a Number to Adjust the Average: A Mathematical Analysis

In this article, we delve into a specific mathematical problem where we analyze the replacement of a number in a set of five numbers to adjust the average. We will explore the underlying mathematical concepts and provide a detailed solution to the problem.

Introduction to the Problem

Consider a set of five numbers, where the average is 60. This means that the sum of these five numbers is 300. If the smallest number in the set is replaced by 80, the new average becomes 65. The objective is to determine the original value of the smallest number that was replaced.

Mathematical Formulation

To formalize the problem, let us denote the five numbers in ascending order as ( u_1 leq u_2 leq u_3 leq u_4 leq u_5 ). The initial average condition can be written as:

Equation 1

( frac{1}{5} cdot (u_1 u_2 u_3 u_4 u_5) 60 )

After replacing the smallest number ( u_1 ) with 80, the new average is 65:

Equation 2

( frac{1}{5} cdot (80 u_2 u_3 u_4 u_5) 65 )

Solving for the Replaced Number

We can now use these two equations to find the value of ( u_1 ).

Step-by-Step Solution

Step 1: Multiply both sides of Equation 1 by 5 to eliminate the fraction:

( u_1 u_2 u_3 u_4 u_5 300 )

Step 2: Multiply both sides of Equation 2 by 5:

( 80 u_2 u_3 u_4 u_5 325 )

Step 3: Subtract Equation 1 from Equation 2 to find ( u_1 ):

( 80 u_2 u_3 u_4 u_5 - (u_1 u_2 u_3 u_4 u_5) 325 - 300 )

( 80 - u_1 25 )

( u_1 80 - 25 55 )

Therefore, the smallest number that was replaced by 80 is 55.

Verification

To verify our solution, we can substitute the values back into the original conditions:

The initial sum of the numbers is 300. If we replace the smallest number, 55, with 80, the new sum is:

( 300 - 55 80 325 )

The new average of the numbers is:

( frac{325}{5} 65 )

This confirms that our solution is correct.

Conclusion

In this article, we have systematically solved a problem where a number in a set of five numbers is replaced to adjust the average. By utilizing basic algebraic manipulation, we were able to determine the original value of the smallest number that was replaced.

The key takeaways from this analysis are:

Understanding the problem: Identify and define the given conditions and the goal of the problem. Formulating the equations: Use algebraic expressions to represent the given conditions. Manipulating equations: Solve for the unknown variable by manipulating and simplifying the equations.

This type of problem-solving can be applied to various numerical and mathematical contexts, enhancing analytical and problem-solving skills essential in fields such as mathematics, statistics, and data science.