Unraveling the Original Price: A Mathematical Puzzle

Mathematics often serves as a tool to solve real-world problems, and here we present a problem that is a great example of the application of algebraic principles. Consider a situation where the price of an item has been changed through a series of percentage adjustments. Specifically, the price has been increased by 25% and then decreased by 25%, resulting in a final price of Rs. 375. The challenge is to determine the original price of the item. This article will explore the step-by-step process to solve this puzzle and provide a detailed explanation of the mathematical reasoning involved.

Understanding the Problem

Let's denote the original price of the item as x. The problem states that after the price of the item is increased by 25%, and then decreased by 25%, the final price is Rs. 375.

Step 1: Increasing the Price by 25%

When the price is increased by 25%, the new price can be calculated as:

Adjusted Price x 25%x x(1 25/100) x(1.25)

Step 2: Decreasing the Price by 25%

The new price is then decreased by 25%. The final price can be calculated as:

Final Price Adjusted Price - 25%(Adjusted Price) Adjusted Price (1 - 25/100) 1.25x (0.75) 0.9375x

According to the problem, the final price is Rs. 375. Therefore, we set up the equation:

0.9375x 375

Step 3: Solving for the Original Price

To find the original price, we solve for x in the equation:

x 375 / 0.9375 400

Thus, the original price of the item is Rs. 400.

Alternative Methods to Solve the Equation

Let's provide some alternative methods to solve this problem:

1. Direct Algebraic Method

Using the variables and steps as described:

x1.250.75 375 x 400

2. Simplified Approach

We can simplify the problem by setting the original price as x. Then, we follow the same steps as above:

x increased by 25% x10025 x1025 125x
x decreased by 25% 125x10075 075125x 09375x
Since the final price is 375, we set up the equation: 09375x 375 and solve for x:
x 375 / 0.9375 400

3. Simplified Steps

Another approach is to set the original price as x. Increase it by 25% and then decrease by 25%:

x increased by 25% 1.25x
Decreased by 25% of 1.25x 0.75(1.25x) 0.9375x
Since 0.9375x 375, solve for x:
x 375 / 0.9375 400

Conclusion

Through these methods, we have consistently arrived at the same conclusion: the original price of the item is Rs. 400. Understanding the underlying principles and steps can help in solving similar mathematical puzzles and increase your proficiency in algebraic problem-solving.

Key Takeaways

Percentage increase and decrease can be expressed as multiplication by fractions. Solving for the original value requires dividing the final value by the compounded percentage change. Algebraic manipulation and equation solving are powerful tools for solving real-world problems.

Related Keywords

Original price, Percentage increase, Percentage decrease, Algebraic solution