Solving The Mobile Merchant's Puzzle: Finding The Cost Of The Lower Priced Mobile
Introduction: The Merchant's Mobile Puzzle
In the realm of mathematical puzzles, we often encounter scenarios that challenge our understanding of percentages, profit, and loss. One such intriguing problem involves a merchant who purchases two mobiles for a combined cost and sells them at varying profit and loss margins. This article delves into a detailed solution of this problem, breaking down each step to ensure clarity and comprehension. We will explore how to effectively apply the concepts of cost price, selling price, profit percentage, and loss percentage to arrive at the correct answer. This problem not only tests our mathematical skills but also our ability to analyze and interpret real-world scenarios involving financial transactions. By understanding the underlying principles and methods, we can confidently tackle similar challenges and enhance our problem-solving capabilities. The key to solving this mobile merchant's dilemma lies in carefully analyzing the given information and applying the appropriate formulas to calculate the cost prices of the individual mobiles.
Problem Statement: Unraveling the Details
The problem presents a scenario where a merchant buys two mobiles for a total cost of Rs. 480. He then sells one mobile at a loss of 15% and the other at a gain of 19%. Crucially, the selling price of both mobiles is the same. Our objective is to determine the cost of the lower-priced mobile. This problem requires us to work backward from the selling prices, which are equal, to find the original cost prices. To solve this, we need to understand the relationship between cost price, selling price, profit, and loss. The loss is calculated as a percentage of the cost price and is subtracted from the cost price to get the selling price. Conversely, the profit is calculated as a percentage of the cost price and is added to the cost price to arrive at the selling price. The problem's complexity arises from the fact that we have two different percentages (15% loss and 19% gain) and a common selling price. This means we need to set up equations that reflect these conditions and solve for the unknowns, which are the individual cost prices of the mobiles. By systematically applying these concepts and equations, we can unravel the details and find the solution to this intriguing mathematical puzzle.
Setting Up the Equations: A Step-by-Step Approach
To effectively solve this problem, let's break down the process into manageable steps. First, we'll denote the cost price of the first mobile as 'x' and the cost price of the second mobile as 'y'. We know that the total cost of both mobiles is Rs. 480, which gives us our first equation:
x + y = 480
Next, we need to consider the profit and loss percentages. The first mobile is sold at a loss of 15%, so its selling price (SP1) can be calculated as:
SP1 = x - 0.15x = 0.85x
The second mobile is sold at a gain of 19%, so its selling price (SP2) can be calculated as:
SP2 = y + 0.19y = 1.19y
According to the problem, the selling prices of both mobiles are equal. This gives us our second equation:
0. 85x = 1.19y
Now we have a system of two equations with two variables (x and y). We can solve this system using various methods, such as substitution or elimination. This systematic approach allows us to translate the word problem into mathematical expressions, making it easier to find the solution. The key is to carefully define the variables and accurately represent the given information in the form of equations. By setting up these equations correctly, we lay the foundation for solving the problem and determining the cost prices of the mobiles.
Solving the Equations: Finding the Cost Prices
Now that we have our system of equations:
- x + y = 480
- 85x = 1.19y
Let's solve for x and y. We can start by solving the first equation for one variable in terms of the other. Let's solve for x:
x = 480 - y
Now, substitute this expression for x into the second equation:
0. 85(480 - y) = 1.19y
Expand and simplify:
408 - 0.85y = 1.19y
Combine the y terms:
408 = 1.19y + 0.85y
408 = 2.04y
Now, solve for y:
y = 408 / 2.04
y = 200
So, the cost price of the second mobile is Rs. 200. Now we can find the cost price of the first mobile by substituting y back into the equation x = 480 - y:
x = 480 - 200
x = 280
Therefore, the cost price of the first mobile is Rs. 280. By carefully substituting and simplifying, we have successfully found the cost prices of both mobiles. This step highlights the importance of algebraic manipulation in solving real-world problems. The ability to solve systems of equations is a valuable skill in various fields, including mathematics, finance, and engineering. With these cost prices, we can now identify the lower-priced mobile and finalize our solution.
Identifying the Lower Priced Mobile: The Final Step
After solving the equations, we found that the cost price of the first mobile (x) is Rs. 280, and the cost price of the second mobile (y) is Rs. 200. To identify the lower-priced mobile, we simply compare these two values.
Since 200 is less than 280, the lower-priced mobile is the second mobile, which cost the merchant Rs. 200. This final step is crucial as it directly answers the question posed in the problem. It demonstrates the importance of not only performing the calculations correctly but also interpreting the results in the context of the problem. In this case, the question specifically asked for the cost of the lower-priced mobile, so we had to identify the smaller of the two calculated cost prices. This step emphasizes the practical application of mathematical problem-solving in real-world scenarios. By correctly identifying the lower-priced mobile, we complete the puzzle and provide a definitive answer to the merchant's dilemma.
Conclusion: The Answer Revealed
In conclusion, by carefully analyzing the problem, setting up the appropriate equations, and solving them systematically, we have successfully determined the cost of the lower-priced mobile. The merchant bought two mobiles for a total of Rs. 480, sold one at a loss of 15%, and the other at a gain of 19%, with both selling for the same price. Our calculations revealed that the cost prices of the two mobiles were Rs. 280 and Rs. 200. Therefore, the cost of the lower-priced mobile is Rs. 200. This problem illustrates the practical application of mathematical concepts such as percentages, profit, loss, and solving systems of equations. It highlights the importance of breaking down complex problems into smaller, manageable steps and applying the correct formulas and techniques. The process of solving this problem not only enhances our mathematical skills but also improves our ability to think critically and solve real-world financial scenarios. The final answer, Rs. 200, represents the culmination of our efforts and demonstrates the power of mathematical reasoning in unraveling complex situations. This exercise underscores the value of mathematics in everyday life and its role in helping us make informed decisions.
By following this comprehensive approach, anyone can tackle similar mathematical challenges with confidence and accuracy.
