Problem 83 - Algebra and Trigonometry

Question

After a \(60 \%\) price reduction, you purchase a computer for \(\$ 440 .\) What was the computer's price before the reduction?

Step-by-Step Solution

Answer

The computer's price before the reduction was $1100.

Step 1: Define the Percentage Relationship

Understand the a reduction of 60% leaves us with 40% of the original value, which in this case is $440.

Step 2: Compute for original price

We know that $440 is 40% of the original price. So, if we denote the original price by \(P\), we can write the equation \(0.4 * P = 440\). Solving for \(P\) gives \(P = 440 / 0.4 \).

Step 3: Solve for P

Calculating the above expression we find that \(P = 1100\).

Key Concepts

Understanding Price Reduction

Price reduction is a common business strategy to attract customers. In the context of percentage problems, a price reduction is essentially a discount applied to the original price of a product. For example, when a store offers a 60% discount, they are reducing the original price by that percentage. This means you only pay the remaining percentage of the product's original price.
To better understand this:

• If a product is originally priced at $100 and there is a 60% reduction, the discount amount is $60.
• The reduced price you pay would be the original price minus the discount, leaving you with $40 to pay.

Understanding how price reduction works helps in setting realistic budgets and allows consumers to take full advantage of sales.

Calculating the Original Price

Finding the original price before a percentage reduction involves understanding what portion of the original price remains after the discount. In this exercise, you know that after a 60% price reduction, you're paying \(440, which represents 40% of the original price.
The steps to calculate the original price can be outlined as:

• Recognize that the amount paid (\)440) is the remaining 40% of the original price.
• Set up the equation: \(0.4 \times P = 440\) where \(P\) represents the original price.
• Solve for \(P\) by dividing \(440\) by \(0.4\): \(P = \frac{440}{0.4}\)
• Calculate the original price: \(P = 1100\)

Now you know that the original price of the computer was $1100 before the discount was applied.

Using Algebraic Equations to Solve Percentage Problems

Algebra plays a crucial role in solving percentage problems, particularly those involving price reductions and original price calculations.
When dealing with these problems, you'll often need to set up an equation that relates the known quantities to the unknown quantities. The key is to express percentages as decimals and translate the percentage problem into a straightforward algebraic equation. For instance:

• If you know the reduced price and the percentage it represents, use the relationship: \(\text{percent remaining} \times \text{original price} = \text{reduced price}\)
• By isolating the original price variable, you solve for it using basic algebraic manipulation, such as division or multiplication.

Algebraic equations simplify these types of problems and ensure you have an accurate calculation, making them indispensable when working with financial and mathematical scenarios.