Solving For 3X Analyzing Income And Expenses

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In this intricate financial puzzle, we embark on a journey to decipher Mukesh's income and expenditures, ultimately aiming to determine the value of '3X'. This problem delves into the realms of percentages, deductions, and remaining balances, requiring a systematic approach to unravel its complexities. Let's dissect the information provided and embark on a step-by-step solution.

Understanding the Financial Landscape

Mukesh's income serves as the foundation of our analysis. We are told that his income is represented by the expression Rs. (X + 500). This means that his income comprises a fixed component of Rs. 500 and a variable component denoted by 'X'. To determine his exact income, we must first find the value of 'X'.

Mukesh's expenses are categorized into two primary areas: rent and travelling. He allocates 32% of his total income towards rent, signifying a significant portion of his earnings. After accounting for the rent expenditure, Mukesh is left with a remaining balance, a portion of which he then spends on travelling. Specifically, he dedicates 25% of the remaining amount to cover his travel expenses.

The remaining amount holds the key to unlocking the value of 'X'. We are informed that after deducting both rent and travel expenses, Mukesh has Rs. 1530 remaining. This figure represents the culmination of all financial transactions and serves as the cornerstone for our calculations.

Formulating the Equation

To determine the value of 'X', we must construct an equation that accurately reflects the financial scenario. Let's break down the process:

  1. Rent Expenditure: Mukesh spends 32% of his income on rent. This translates to (32/100) * (X + 500).

  2. Remaining Amount after Rent: After paying rent, Mukesh is left with (100% - 32%) = 68% of his income. This can be expressed as (68/100) * (X + 500).

  3. Travelling Expenditure: Mukesh spends 25% of the remaining amount on travelling. This equates to (25/100) * (68/100) * (X + 500).

  4. Final Remaining Amount: After deducting both rent and travel expenses, Mukesh has Rs. 1530 remaining. This can be represented as:

    (68/100) * (X + 500) - (25/100) * (68/100) * (X + 500) = 1530

This equation forms the crux of our solution. By solving for 'X', we can unravel the unknown variable and gain insights into Mukesh's financial standing.

Solving for X: A Step-by-Step Approach

Now that we have formulated the equation, let's embark on the journey of solving for 'X'. This involves a series of algebraic manipulations, each step bringing us closer to the solution.

  1. Simplifying the Equation: The initial equation can be simplified by combining like terms and reducing fractions. This yields:

    (0.68) * (X + 500) - (0.25) * (0.68) * (X + 500) = 1530

  2. Factoring out (X + 500): To further simplify the equation, we can factor out the common term (X + 500):

    (X + 500) * (0.68 - 0.25 * 0.68) = 1530

  3. Calculating the Coefficient: Evaluating the expression within the parentheses, we get:

    (X + 500) * (0.68 - 0.17) = 1530

    (X + 500) * (0.51) = 1530

  4. Isolating (X + 500): To isolate the term (X + 500), we divide both sides of the equation by 0.51:

    X + 500 = 1530 / 0.51

    X + 500 = 3000

  5. Solving for X: Finally, to solve for 'X', we subtract 500 from both sides of the equation:

    X = 3000 - 500

    X = 2500

Therefore, the value of 'X' is determined to be Rs. 2500. This represents the variable component of Mukesh's income.

Determining the Value of 3X

With the value of 'X' firmly established, we can now proceed to calculate the value of '3X'. This is a straightforward multiplication operation:

3X = 3 * 2500

3X = 7500

Thus, the value of '3X' is Rs. 7500. This is the ultimate answer we sought to uncover.

Verifying the Solution: A Comprehensive Check

To ensure the accuracy of our solution, let's perform a comprehensive check by plugging the value of X back into the original problem statement and verifying if all conditions are satisfied.

  1. Mukesh's Income: Mukesh's income is given by (X + 500). Substituting X = 2500, we get:

    Income = 2500 + 500 = Rs. 3000

  2. Rent Expenditure: Mukesh spends 32% of his income on rent. This amounts to:

    Rent = (32/100) * 3000 = Rs. 960

  3. Remaining Amount after Rent: After paying rent, Mukesh is left with:

    Remaining Amount = 3000 - 960 = Rs. 2040

  4. Travelling Expenditure: Mukesh spends 25% of the remaining amount on travelling. This equates to:

    Travel Expenses = (25/100) * 2040 = Rs. 510

  5. Final Remaining Amount: After deducting both rent and travel expenses, Mukesh has:

    Final Remaining Amount = 2040 - 510 = Rs. 1530

This matches the information provided in the problem statement, confirming the validity of our solution. We have successfully verified that the value of 3X is indeed Rs. 7500.

Conclusion: A Triumph of Financial Acumen

In this intricate financial analysis, we have successfully navigated the complexities of Mukesh's income and expenditures. By formulating an equation, systematically solving for 'X', and meticulously verifying our solution, we have arrived at the conclusive answer: the value of '3X' is Rs. 7500. This problem serves as a testament to the power of mathematical principles in deciphering real-world financial scenarios. Understanding the intricacies of income, expenses, and remaining balances empowers us to make informed financial decisions and achieve our economic goals.

This exercise underscores the importance of careful analysis, systematic problem-solving, and meticulous verification in the realm of finance. By mastering these skills, we can confidently tackle financial challenges and pave the way for a secure and prosperous future.