Calculate Original Bill Amount If 30 Rupees Due After Paying 4/5

In the realm of mathematical puzzles, real-world scenarios often provide the most engaging challenges. Today, we embark on a journey to decipher a bill payment conundrum. The problem states that after paying 4/5 of a bill, a balance of ₹30 remains outstanding. Our mission is to determine the original amount of the bill. This is a classic problem that combines fractions and basic algebra, making it an excellent exercise for honing our mathematical skills. To effectively tackle this problem, we need to break it down into manageable steps. First, we must understand the fractional representation of the payment. If 4/5 of the bill has been paid, then 1/5 of the bill remains unpaid. This is a crucial piece of information because it directly relates the unpaid portion to the remaining balance of ₹30. From here, we can set up a simple equation to solve for the total amount of the bill. Let's denote the total bill amount as 'x'. According to the problem, 1/5 of x is equal to ₹30. This translates into the equation (1/5)x = 30. To solve for x, we need to isolate the variable by multiplying both sides of the equation by 5. This gives us x = 30 * 5, which simplifies to x = 150. Therefore, the original amount of the bill was ₹150. This methodical approach, breaking the problem into smaller, logical steps, highlights the power of mathematics in solving everyday financial questions.

Setting Up the Equation

Understanding the problem requires translating the word problem into a mathematical equation. This is a core skill in mathematics, as it allows us to apply algebraic techniques to solve real-world scenarios. In our case, the statement “4/5 of a bill is paid, and ₹30 is still due” forms the basis of our equation. We need to represent the unknown bill amount with a variable. Let's use 'x' to denote the total amount of the bill. The problem tells us that 4/5 of the bill has been paid. This means that 1/5 of the bill is still outstanding. We know that this outstanding amount is equal to ₹30. Therefore, we can write the equation as (1/5) * x = 30. This equation is the key to unlocking the solution. It directly relates the fraction of the bill that is unpaid to the remaining balance. To solve for 'x', we need to isolate the variable. This involves performing the same operation on both sides of the equation to maintain the balance. In this case, we need to multiply both sides of the equation by 5. This will cancel out the fraction on the left side and leave us with the value of 'x'. The equation (1/5) * x = 30 transforms into x = 30 * 5. This simplification is a fundamental algebraic step, demonstrating the power of inverse operations. By multiplying both sides by 5, we effectively undo the division by 5, isolating 'x' and revealing the total bill amount. This clear and concise equation setup is the cornerstone of solving this mathematical problem.

Solving for the Total Bill Amount

With the equation (1/5) * x = 30 established, the next step is to solve for 'x', which represents the total bill amount. Solving equations is a fundamental skill in algebra, and it involves isolating the variable to determine its value. In this case, 'x' is being multiplied by 1/5. To isolate 'x', we need to perform the inverse operation, which is multiplying both sides of the equation by 5. This will cancel out the fraction on the left side and leave 'x' by itself. Multiplying both sides of the equation by 5, we get: 5 * (1/5) * x = 5 * 30. On the left side, the 5 and 1/5 cancel each other out, leaving us with just 'x'. On the right side, 5 * 30 equals 150. Therefore, the equation simplifies to x = 150. This means that the total bill amount was ₹150. The process of solving for 'x' demonstrates a core principle of algebra: performing the same operation on both sides of an equation maintains the equality. By multiplying both sides by 5, we effectively undid the division by 5, isolating 'x' and revealing its value. This methodical approach is crucial for solving algebraic equations accurately. Now that we have determined the total bill amount, we can verify our answer by plugging it back into the original problem. If the total bill was ₹150, then 4/5 of the bill would be (4/5) * 150 = ₹120. This means that ₹150 - ₹120 = ₹30 would still be due, which matches the information given in the problem.

Verifying the Solution

After arriving at a solution in any mathematical problem, it's crucial to verify its accuracy. This step ensures that our calculations are correct and that our answer aligns with the problem's conditions. In our bill payment scenario, we've determined that the total bill amount was ₹150. To verify this, we need to check if paying 4/5 of the bill leaves a balance of ₹30. First, let's calculate 4/5 of ₹150. This can be done by multiplying 150 by 4/5: (4/5) * 150 = (4 * 150) / 5 = 600 / 5 = ₹120. This means that 4/5 of the bill is ₹120. Now, let's subtract this amount from the total bill amount to find the remaining balance: ₹150 - ₹120 = ₹30. The remaining balance is indeed ₹30, which matches the information given in the problem. This confirms that our solution of ₹150 for the total bill amount is correct. Verification is a vital step in problem-solving. It not only confirms the accuracy of our answer but also reinforces our understanding of the problem and the steps taken to solve it. By plugging our solution back into the original problem, we ensure that all conditions are met and that our answer is logical and consistent. This process builds confidence in our mathematical abilities and helps us avoid errors in future problem-solving endeavors. In conclusion, the verification step solidifies our understanding and ensures the reliability of our solution.

Conclusion: The Bill Amount Revealed

In summary, we successfully navigated through the bill payment problem, utilizing a combination of fractional understanding and algebraic techniques. We began by deciphering the problem statement, recognizing that the unpaid 1/5 of the bill corresponded to the remaining balance of ₹30. This crucial insight allowed us to formulate the equation (1/5) * x = 30, where 'x' represented the total bill amount. Solving this equation involved multiplying both sides by 5, effectively isolating 'x' and revealing its value as ₹150. This methodical approach demonstrated the power of algebraic manipulation in solving real-world problems. To ensure the accuracy of our solution, we performed a verification step. We calculated 4/5 of ₹150, which equaled ₹120, and then subtracted this amount from the total bill amount. The resulting balance of ₹30 perfectly matched the problem's conditions, confirming the correctness of our answer. This verification process highlighted the importance of checking our work to avoid errors and build confidence in our solutions. This problem serves as a valuable exercise in applying mathematical concepts to practical scenarios. It reinforces our understanding of fractions, algebra, and the problem-solving process. By breaking down the problem into smaller, manageable steps, we were able to systematically arrive at the solution. The final answer, ₹150, represents the total amount of the bill, effectively resolving the mathematical puzzle.

Keywords: bill amount, mathematical equation, algebra, fractions, problem-solving.