Problem 52 - College Algebra

Question

After a graphing calculator's price is reduced by \(\frac{1}{3}\) of its original price, you purchase it for \(\$ 64 .\) What was the graphing calculator's price before the reduction?

Step-by-Step Solution

Answer

The graphing calculator's original price before the reduction was $96.

Step 1: Understand the Reduction

Firstly, comprehend the reduction. When the price of a product is reduced by \(\frac{1}{3}\), it means only \(\frac{2}{3}\) of its original price remains.

Step 2: Set Up an Equation

Next, it is known that the remaining \(\frac{2}{3}\) of the original price is equal to $64. This can be expressed as an equation: \(\frac{2}{3} * original\_price = $64.\)

Step 3: Solve for the Original Price

To find the original price, the equation can be solved by dividing both sides by \(\frac{2}{3}\). In other words, original\_price = $\frac{64} {\frac{2}{3}} = $96

Key Concepts

Solving Equations

Equations are like mathematical puzzles. Solving them means finding the value of the unknown variable that makes the equation true. In our exercise, the unknown variable is the original price of the calculator.

The equation given is \(\frac{2}{3} \times \text{original\_price} = 64\). This type of equation arises often when dealing with parts of a whole. The goal is to solve for "original\_price."

• First, understand that the equation tells you that 2/3 of the original price equates to 64.
• To isolate the variable (original\_price), you need to "cancel out" the fraction.

This step involves dividing both sides by \(\frac{2}{3}\), which is the coefficient of the original price. Performing this division is like reversing the multiplication: \[\text{original\_price} = \frac{64}{\frac{2}{3}} = 64 \times \frac{3}{2} = 96\]This tells us that the calculator originally cost $96 before the price reduction. Equation solving requires manipulating both sides of the equation to "balance" it until the variable is on one side alone.

Price Reduction

The concept of price reduction is a common aspect of everyday situations, like sales and discounts. In the exercise, it's given that the price of the calculator was reduced by 1/3 of its original price.

Here's how it works in simple terms:

• When a price is reduced by a fraction, like 1/3, it means you're subtracting that fraction from the whole.
• The remaining part of the price is what you actually pay, which in this case is 2/3 of the original price.

Understanding this process is crucial because it helps in knowing how much of the original cost you are saving and how much you still need to pay. Knowing how to calculate this can aid in budgeting and shopping efficiently without over-spending or being "fooled" by seemingly attractive discounts.

Proportional Reasoning

Proportional reasoning involves recognizing relationships between quantities. When two ratios are equal, you are dealing with proportions. This is essential for solving many real-world problems.

In our calculator example, the price reduction is a proportional reasoning challenge:

• The starting point is the original price being 100% before any discount.
• When the calculator’s price is reduced, it effectively proportions this as a new part-whole relationship between the actual paid amount ($64) and the unknown original price.

The understanding here is that the "whole" is now described as "2/3 of the original," not as a different fixed value. Mastering proportional reasoning enables you to decode complex problems by understanding how parts relate to a whole and applying simple mathematics to find solutions, like determining that the original price was $96 in this case.