Calculating Original Price Of A Computer After A 15% Discount
In the realm of mathematics, particularly in practical applications such as calculating discounts and original prices, a clear understanding of percentages and algebraic equations is essential. This article delves into a specific problem: determining the original price of a computer given its sale price and the percentage discount applied. We'll explore the underlying mathematical principles and provide a step-by-step solution, ensuring a comprehensive understanding of the process.
Understanding the Problem: Sale Price and Discount
At the heart of our problem lies the relationship between the original price, the discount percentage, and the sale price. Let's break down these key elements:
- Original Price: This is the initial price of the computer before any discount is applied. It's the value we aim to find.
- Discount Percentage: This is the percentage by which the original price is reduced. In our case, it's 15%.
- Sale Price: This is the price of the computer after the discount is applied. We are given that the sale price is $2550.
The key here is to recognize that the sale price represents a portion of the original price. If a 15% discount is applied, the sale price represents 100% - 15% = 85% of the original price. This understanding forms the basis for our mathematical approach.
Setting up the Equation: A Percentage Problem
To solve for the original price, we need to translate the problem into a mathematical equation. Let's represent the original price with the variable 'x'. We know that 85% of the original price (x) equals the sale price ($2550). We can write this as:
- 85 * x = $2550
This equation expresses the fundamental relationship between the original price, the discount, and the sale price. The decimal 0.85 represents 85%, which is the percentage of the original price that the sale price constitutes. Solving this equation will give us the original price of the computer.
Solving for the Original Price: A Step-by-Step Approach
Now that we have our equation, 0. 85 * x = $2550, we can solve for 'x', the original price. To do this, we need to isolate 'x' on one side of the equation. We can achieve this by dividing both sides of the equation by 0.85:
x = $2550 / 0.85
Performing this division will give us the value of 'x', which is the original price of the computer.
Calculation and Result
Let's perform the calculation:
x = $2550 / 0.85 = $3000
Therefore, the original price of the computer was $3000.
Verifying the Solution: Ensuring Accuracy
To ensure our solution is correct, we can verify it by calculating the discount amount and subtracting it from the original price to see if it matches the sale price.
The discount amount is 15% of the original price, which is 0.15 * $3000 = $450.
Subtracting the discount from the original price, we get $3000 - $450 = $2550, which matches the given sale price. This confirms that our solution of $3000 for the original price is correct.
Alternative Approaches: Proportions and Ratios
While we used a direct algebraic equation, there are alternative approaches to solving this problem, such as using proportions or ratios.
Using Proportions
We can set up a proportion to relate the sale price to the original price. A proportion is an equation stating that two ratios are equal. In this case, we can say:
Sale Price / Original Price = 85 / 100
Plugging in the sale price ($2550) and representing the original price as 'x', we get:
$2550 / x = 85 / 100
To solve this proportion, we can cross-multiply:
$2550 * 100 = 85 * x
$255000 = 85x
Dividing both sides by 85, we get:
x = $3000
This confirms our previous result.
Using Ratios
We can also use ratios to solve the problem. The ratio of the sale price to the original price is 85:100, which can be simplified to 17:20. We can set up the ratio:
$2550 : x = 17 : 20
This means that $2550 corresponds to 17 parts, and we need to find the value of 20 parts. To do this, we can find the value of one part by dividing $2550 by 17:
One part = $2550 / 17 = $150
Then, we multiply the value of one part by 20 to find the original price:
Original price = $150 * 20 = $3000
Again, this confirms our original solution.
Real-World Applications: Discounts and Pricing
Understanding how to calculate original prices from sale prices and discounts has numerous real-world applications. From retail shopping to financial planning, these skills are invaluable. Here are a few examples:
- Retail Shopping: When evaluating sales and discounts, knowing how to calculate the original price helps you determine the true value of the discount and make informed purchasing decisions. You can assess whether a sale price is genuinely a good deal or if the item is still overpriced compared to its original value.
- Budgeting and Finance: Understanding discounts and pricing is crucial for effective budgeting. By calculating the actual savings from discounts, you can track your spending and make informed financial decisions. This is particularly useful when making significant purchases or planning for long-term expenses.
- Business and Sales: For businesses, understanding pricing strategies and discounts is essential for attracting customers and maximizing profits. Calculating discounts, markdowns, and original prices accurately ensures that pricing decisions are sound and profitable. This includes setting competitive prices, managing inventory, and running effective sales promotions.
- Real Estate: In real estate, understanding pricing and discounts is vital for both buyers and sellers. Buyers can calculate the original price of a property to assess its value and negotiate effectively. Sellers can determine appropriate listing prices and offer discounts strategically to attract potential buyers.
Common Mistakes: Avoiding Pitfalls
While the calculation itself is straightforward, there are common mistakes people make when dealing with percentage problems. Being aware of these pitfalls can help you avoid errors and arrive at the correct solution.
- Misinterpreting the Percentage: One common mistake is misunderstanding what percentage the sale price represents. In our case, the sale price is 85% of the original price, not 15%. Failing to recognize this can lead to incorrect calculations. Always clarify what the percentage refers to – is it a percentage of the original price or a percentage of the sale price?
- Incorrectly Setting up the Equation: Setting up the equation incorrectly is another common mistake. For example, some might mistakenly calculate 15% of the sale price and add it to the sale price. This will not give the original price. The equation must accurately reflect the relationship between the original price, discount, and sale price.
- Arithmetic Errors: Simple arithmetic errors can also lead to incorrect answers. Double-check your calculations, especially when dealing with decimals and divisions. Using a calculator can help reduce the likelihood of these errors.
- Not Verifying the Solution: It's always a good practice to verify your solution. By calculating the discount amount and subtracting it from the original price, you can confirm whether the result matches the given sale price. This step helps catch any errors in your calculations and ensures the accuracy of your answer.
Conclusion: Mastering Percentage Problems
Calculating the original price from a sale price and discount is a fundamental mathematical skill with wide-ranging applications. By understanding the relationship between percentages, original prices, and discounts, you can solve various real-world problems, from shopping deals to financial planning. In this article, we explored a step-by-step approach to solving this type of problem, discussed alternative methods like proportions and ratios, and highlighted common mistakes to avoid. Mastering these skills enhances your mathematical proficiency and equips you to make informed decisions in diverse situations.
The original price of the computer, in this case, was $3000. This process demonstrates the power of mathematics in everyday scenarios and the importance of accurate calculations in financial contexts.
By grasping these concepts and practicing similar problems, you can strengthen your understanding of percentage-related calculations and confidently tackle real-world challenges involving discounts and pricing.