Rama's Weekly Salary A Mathematical Problem Solving

In this article, we'll dissect a common financial puzzle involving fractions and basic algebra. The problem at hand centers around Rama's weekly salary and how she allocates it among rent, food, and other expenses. Our mission is to determine Rama's total weekly income, given the fractional expenditures and the remaining amount.

The Salary Predicament

The core of the problem states that Rama spends 5/8 of her salary on rent, and then allocates 1/3 of the remaining salary to food. After these expenses, she has Rs. 40 left for other expenditures. This scenario presents a classic fractional problem often encountered in mathematics and real-life budgeting situations. To solve this, we'll employ algebraic principles, turning the word problem into a mathematical equation.

Setting Up the Algebraic Framework

To demystify the problem, let's denote Rama's weekly salary as 'X'. This 'X' is the variable we aim to find. The first key piece of information is that Rama spends 5/8 of her salary on rent. This translates to (5/8) * X spent on rent. Consequently, the remaining salary after paying rent is X - (5/8) * X. This expression represents the portion of Rama's salary that isn't yet allocated.

Navigating the Remaining Salary

Following the rent payment, Rama spends 1/3 of the remaining salary on food. This is a crucial step, as we're not calculating 1/3 of the total salary, but 1/3 of what's left after the rent deduction. Mathematically, this is expressed as (1/3) * [X - (5/8) * X]. This portion represents the amount spent on food.

The Final Equation The Path to the Solution

After accounting for rent and food expenses, Rama has Rs. 40 remaining. This gives us the final piece of the puzzle the equation. We can express the situation as: X - (5/8) * X - (1/3) * [X - (5/8) * X] = 40. This equation encapsulates the entire problem, and solving it will reveal Rama's weekly salary. The left side represents the total salary minus the amounts spent on rent and food, which equals the remaining Rs. 40.

Now, let's embark on the journey of solving the equation we've established: X - (5/8) * X - (1/3) * [X - (5/8) * X] = 40. This involves simplifying, combining like terms, and isolating 'X' to find its value. Each step is a careful manipulation of the equation to bring us closer to the solution.

Simplifying the Expression

The initial step is to simplify the expression within the brackets. We have X - (5/8) * X inside the brackets. To combine these terms, we need a common denominator. We can rewrite 'X' as (8/8) * X. Thus, the expression becomes (8/8) * X - (5/8) * X, which simplifies to (3/8) * X. This simplification makes the equation more manageable.

Distributing and Combining Like Terms

Next, we substitute the simplified bracketed expression back into the equation: X - (5/8) * X - (1/3) * (3/8) * X = 40. Now, we multiply (1/3) by (3/8) * X, resulting in (1/8) * X. The equation now reads: X - (5/8) * X - (1/8) * X = 40. To further simplify, we combine the terms containing 'X'. We have X, -(5/8) * X, and -(1/8) * X. To combine these, we need a common denominator, which is 8. We rewrite 'X' as (8/8) * X. The equation becomes (8/8) * X - (5/8) * X - (1/8) * X = 40. Combining these terms gives us (2/8) * X or (1/4) * X.

Isolating 'X' The Final Calculation

Our equation is now simplified to (1/4) * X = 40. To isolate 'X', we need to get rid of the (1/4) coefficient. We can do this by multiplying both sides of the equation by 4. This gives us X = 40 * 4. Performing this multiplication, we find that X = 160. This is the solution to our equation and represents Rama's weekly salary.

Having determined that Rama's weekly salary is Rs. 160, it's insightful to break down her budget to understand how her money is allocated. This involves calculating the amounts she spends on rent and food based on the fractions given in the problem.

Calculating Rent Expenditure

The problem states that Rama spends 5/8 of her salary on rent. Since her salary is Rs. 160, the amount spent on rent is (5/8) * 160. To calculate this, we multiply 5 by 160, which equals 800, and then divide by 8. This yields a rent expenditure of Rs. 100. This significant portion of her salary highlights the financial commitment of housing expenses.

Determining Food Expenses

After paying rent, Rama spends 1/3 of the remaining salary on food. We know that after paying Rs. 100 for rent from her Rs. 160 salary, she has Rs. 60 remaining. So, the amount spent on food is (1/3) * 60, which equals Rs. 20. This is the amount Rama allocates to food expenses each week.

Verifying the Remaining Amount

To ensure our calculations are correct, let's verify that the remaining amount after rent and food expenses matches the Rs. 40 stated in the problem. Rama's total expenses are Rs. 100 (rent) + Rs. 20 (food) = Rs. 120. Subtracting this from her total salary of Rs. 160, we get Rs. 160 - Rs. 120 = Rs. 40. This matches the amount stated in the problem, confirming the accuracy of our solution.

Through our step-by-step analysis, we've successfully decoded Rama's financial situation. By translating the word problem into an algebraic equation and meticulously solving it, we've arrived at the solution.

Concluding Rama's Salary

The answer to the question,