Problem 30 - Elementary Algebra Within Reach

Question

In Exercises 29-34, use a system of linear equations to solve the problem. The sale price of a surround sound system is \(\$ 716\). The discount is \(20 \%\) of the original price. Find the original price.

Step-by-Step Solution

Answer

The original price of the surround sound system is \$895.

Step 1: Set Up the Equation

First, let's consider the original price as \(X\). Since the sale price is obtained after a discount of \(20\%\), we can represent this situation mathematically as follows: \[X - 0.2X = 716\] This equation represents the sale price (\$716) as the original price (\(X\)) minus \(20\%\) of the original price (\(0.2X\)).

Step 2: Simplify the Equation

The equation in step 1 can be simplified by combining like terms. This will result in:\[0.8X = 716\] which represents that \(80\%\) of the original price is equal to the sale price.

Step 3: Solve for X

Finally, to find the original price (\(X\)), we now solve for \(X\) by dividing both sides of the equation by 0.8:\[X = \frac{716}{0.8}\]We can compute this division to find the value of \(X\).

Key Concepts

Understanding Discount Problems

Discount problems are common scenarios where a specific percentage is reduced from the original price of an item. When faced with a discount problem, it's crucial to understand what the original price is and how the discount affects it. Typically:

• The original price is the starting point, the cost before any reductions.
• a discount represents a percentage taken off from the original price.
• The sale price is what the customer pays after the discount is applied.

In our example, the surround sound system has a sale price of $716 after a 20% discount. This means 20% of the original price was subtracted to arrive at this sale price. Formulating this situation into a mathematical problem helps us find out what the original price was, helping to bridge the understanding gap in typical discount scenarios.

Solving Equations

When dealing with discounts, finding out the original price involves solving a simple linear equation. In such equations, you work to isolate the variable representing the unknown amount, in this case, the original price.To begin, recognize that the sale price equation is derived from:\[ X - 0.2X = 716 \]Here, the original price is represented by \( X \) and the sale price by \( 716 \). Simplifying the equation through combining like terms gives:\[ 0.8X = 716 \]At this step, the equation tells us that the sale price is 80% of the original price. To solve for \( X \), divide both sides by 0.8 to isolate \( X \):\[ X = \frac{716}{0.8} \]This calculation will reveal the original price, showing us the importance of understanding and solving linear equations in discount problems.

Calculating the Sale Price

The sale price is essentially the final price after the original price has undergone the discount. Calculating this involves knowing both the original price and the discount percentage. When the discount is applied:

• First, calculate the discount amount by converting the percentage to a decimal and multiplying it by the original price.
• Subtract this discount amount from the original price to find the sale price.

For example, if the original price is determined to be \( X \), which we calculated to be \( \frac{716}{0.8} \), the discount would be \( 0.2X \). The final step of getting the sale price involves confirming:\[ \text{Sale Price} = X - 0.2X \]This simplification process ensures students understand not only how discounts impact pricing but also the arithmetic used to calculate sale prices from given percentages and original prices.