Josh Earned $72 Less Than His Sister Who Earned $93 More Than Her Mom. If They Earned A Total Of $504,

Josh Earned $72 Less Than His Sister Who Earned $93 More Than Her Mom. If They Earned A Total Of $504, understanding the relationships between their earnings involves unraveling some basic algebra and applying logical reasoning. This problem, while seemingly simple, offers a great opportunity to explore how to set up equations, solve for unknowns, and interpret the results in a real-world context. In this article, we'll dive into the details of this problem, analyze the relationships between the three individuals' earnings, and provide a comprehensive guide on how to approach similar word problems.

Understanding the Problem Statement

The problem involves three individuals: Josh, his sister, and their mom. The key points are:

• Josh earned $72 less than his sister.


• The sister earned $93 more than her mom.


• The total combined earnings of all three individuals amount to $504.

Using these clues, our goal is to determine the individual earnings of Josh, his sister, and their mom.

Breaking Down the Relationships

Before jumping into the calculations, it's important to understand the relationships described:

1. Josh's earnings relative to his sister:

Josh earned $72 less than his sister.
Mathematically:
Josh's earnings = Sister's earnings - 72

1. Sister's earnings relative to her mom:

The sister earned $93 more than her mom.
Mathematically:
Sister's earnings = Mom's earnings + 93

1. Total earnings:

Combined earnings of Josh, his sister, and their mom sum to $504:
Josh + Sister + Mom = 504

Using these relationships, we can set up equations to find each person's earnings.

Setting Up the Equations

Let’s define variables for each individual:

• Let J = Josh's earnings


• Let S = Sister's earnings


• Let M = Mom's earnings

Based on the relationships:

• From point 1:

J = S - 72

• From point 2:

S = M + 93

• The total sum:

J + S + M = 504

Now, substitute the expressions for J and S into the total earnings equation:

1. Replace J with S - 72:

(S - 72) + S + M = 504

1. Replace S with M + 93:

( (M + 93) - 72 ) + (M + 93) + M = 504

Now, simplify the equation.

Solving the Equations Step-by-Step

Let's go through the algebraic steps:

1. Simplify the first term:

(M + 93 - 72) = M + 21

1. Rewrite the entire sum:

(M + 21) + (M + 93) + M = 504

1. Combine like terms:

M + 21 + M + 93 + M = 504

1. Sum the M terms:

(M + M + M) = 3M

1. Sum the constants:

21 + 93 = 114

So, the equation becomes:

3M + 114 = 504

1. Subtract 114 from both sides:

3M = 504 - 114
3M = 390

1. Divide both sides by 3:

M = 390 / 3
M = 130

Now, having found Mom's earnings, find the sister's earnings:

S = M + 93 = 130 + 93 = 223

And Josh's earnings:

J = S - 72 = 223 - 72 = 151

Final Earnings Breakdown

Based on the calculations:

• Mom's earnings: $130


• Sister's earnings: $223


• Josh's earnings: $151

Check the total sum:
151 + 223 + 130 = 504

The total matches the problem statement, confirming our solution.

Interpreting the Results

This problem illustrates essential algebraic techniques, including substitution and solving linear equations, which are fundamental in solving real-world word problems. Understanding the relationships between variables allows us to set up equations effectively and find precise solutions.

Summary of earnings:

• Josh earned $151


• His sister earned $223


• Their mom earned $130

Applications and Similar Problems

This type of problem can be adapted to various scenarios involving relationships between different quantities. Some common applications include:

• Budget allocations among family members or departments


• Salary comparisons within organizations


• Profit sharing models


• Distribution of resources based on certain rules or relationships

Sample similar problems include:

• "A and B's combined age is 60 years. B is 5 years older than A. What are their ages?"


• "A store sells two products with a combined revenue of $1000. Product A earns $200 more than Product B. Find the individual revenues."

Tips for Solving Similar Word Problems

When approaching similar problems, consider these steps:

• Identify the variables: Assign a variable to each unknown quantity.
• Translate relationships into equations: Express the relationships mentioned in the problem using algebra.
• Set up the total sum equation: Combine all variables with their relationships.
• Solve systematically: Use substitution or elimination methods to find variable values.
• Verify your solutions: Check if the solution satisfies all original relationships and the total sum.

Conclusion

Understanding how to interpret relationships in word problems and translating them into equations is a vital skill in mathematics. The problem of calculating individual earnings based on their differences exemplifies how algebra can provide clarity and precision. By carefully setting up equations and solving step-by-step, you can efficiently solve problems involving multiple variables and relationships. Whether in academic settings or real-life situations like budgeting or resource allocation, these skills are invaluable for making informed decisions and analyses.

Remember: Practice makes perfect. The more you practice setting up and solving such problems, the more intuitive they will become, enabling you to tackle even more complex scenarios with confidence.