Salaries and Proportional Relationships: A Mathematical Analysis

In today's workplace, understanding the relationships between salaries can provide valuable insights into financial dynamics. This article delves into a mathematical problem where two colleagues, Joe and Bob, work at the same workplace and have a combined salary of $10,000. However, the relationship between their salaries is expressed as a proportional equation, which leads to the question: Who earns more and by how much?

The Problem Context

Joe and Bob work for the same company and this month they received a combined salary of $10,000. Additionally, we are given that 1/11 of Joe's salary is equal to 1/9 of Bob’s salary. This relationship can be expressed as:

J B 10000 J/11 B/9

Where J is Joe's salary and B is Bob's salary. The objective is to determine who earns more and the difference in their salaries.

Solving the Equations

Let's start by expressing the proportional equation as:

9J 11B

We can then solve for J in terms of B by rearranging the equation:

J 11B/9

Now, we substitute this expression for J into the first equation:

(11B/9) B 10000

To combine the terms, express B with a common denominator:

11B/9 9B/9 10000

Combining the terms:

20B/9 10000

Solving for B by multiplying both sides by 9/20:

B 10000 * 9/20 4500

Now, substitute B 4500 back into the first equation to find J:

J 10000 - 4500 5500

Thus, we find that:

Joe's salary: $5500 Bob's salary: $4500

Conclusion

The analysis clearly indicates that Joe earns more than Bob. The difference in their salaries is as follows:

5500 - 4500 1000

Thus, Joe earns $1000 more than Bob.

Alternative Methods

Let's consider another approach to solve the same problem using the Least Common Multiple (LCM) method:

Find the LCM of 11 and 9: 99 If Joe's salary is 99, then 1/11 of 99 9, and 1/9 of Bob's salary is also 9. Setting up the equations, the sum of their salaries is 99 81 180, which is scaled up to 10,000.

Solving these equations further yields:

Bob's salary: $4500 Joe's salary: $5500

Resulting in the same conclusion that Joe earns $1000 more than Bob.

The Technique of Algebraic Substitution

Finally, let's review the technique of algebraic substitution using a different variable setup:

Let J represent Joe's salary and B represent Bob's salary. The equations are J B 10000 and J/11 B/9. Rearrange the second equation to find J: J 11B/9. Substitute J into the first equation: (11B/9) B 10000. Solving the combined equation: 20B/9 10000 → B 4500. Finally, find J: 11 * 4500 / 9 5500.

This method confirms that Joe earns $5500 and Bob earns $4500, with a difference of $1000.

Conclusion

In this analysis, we have explored multiple methods of solving a problem involving two colleagues' salaries and their proportional relationship. The result remains clear: Joe earns more than Bob, with a difference of $1000. Understanding such relationships is crucial in financial planning and workplace dynamics.