Problem 13 - Algebra 2

Question

For Exercises \(10-13,\) use the following information. Mai-Lin is shopping for computer software. She finds a CD-ROM that costs \(\$ 49.99,\) but is on sale at a 25\(\%\) discount. She also has a \(\$ 5\) coupon she can use. Which method results in the lower sale price? Explain your reasoning.

Step-by-Step Solution

Answer

Apply the 25\(\%\) discount first, then the \$5 coupon for the lowest price of \$32.49.

Step 1: Calculate the Sale Price with Discount First

First, let's find out what happens if Mai-Lin applies the 25\(\%\) discount before using her \(\\(5\) coupon. The original price of the CD-ROM is \(\\)49.99\). To find the discount amount, we calculate:\[\text{Discount} = 49.99 \times 0.25 = 12.4975\]Subtracting this discount from the original price gives:\[49.99 - 12.4975 = 37.4925\]Next, apply the \(\\(5\) coupon:\[37.4925 - 5 = 32.4925\]Thus, the final price using this method is \(\\)32.49\).

Step 2: Calculate the Sale Price with Coupon First

Now, let's examine the scenario where the \(\\(5\) coupon is applied before the 25\(\%\) discount.Starting with the original price of \(\\)49.99\), apply the coupon first:\[49.99 - 5 = 44.99\]Now apply the 25\(\%\) discount to \(\\(44.99\):\[\text{Discount} = 44.99 \times 0.25 = 11.2475\]Subtract this discount from the reduced price:\[44.99 - 11.2475 = 33.7425\]The final price using this method is \(\\)33.74\).

Step 3: Compare the Final Prices

Now that we have calculated the final prices using both methods, let's compare:- Applying the discount first results in a price of \(\\(32.49\).- Applying the coupon first results in a price of \(\\)33.74\).Thus, the lower price is obtained by applying the discount first, followed by the coupon.

Key Concepts

Coupon Application

A coupon is a fantastic way to save money on purchases. In Mai-Lin's case, she has a $5 coupon that she can use to reduce the overall cost of her CD-ROM. Understanding how to apply coupons effectively is crucial since it can significantly impact the final purchase price.

When using a coupon, it is important to recognize its face value. Here, it's worth $5, meaning, it will reduce the bill by that amount. However, the order of application—whether you use the coupon before or after any discounts—is essential in calculating the total savings.

• Using a coupon after a percentage discount reduces the price more significantly, as seen in Mai-Lin’s scenario.
• Applying a coupon first can result in less savings if followed by a percentage discount.

Hence, the strategy in coupon application should consider the sequence of usage for maximum benefit.

Discount Calculation

Discounts are typically offered as a percentage reduction on the original price. In Mai-Lin's case, her selected CD-ROM is offered at a 25% discount. Understanding how to compute discounts simplifies shopping decisions.

To calculate the discount, you multiply the original price by the discount rate, which is 25% for Mai-Lin. Here's how the calculation works:

• First, convert the percentage to a decimal by dividing by 100, which in this case is 0.25.
• Multiply the original price of $49.99 by 0.25 to find the discount amount, which is approximately $12.50.
• The discount is then subtracted from the original price to find the new sale price.

Knowing this basic calculation ensures you can quickly determine how much you will pay after discounts are applied.

Cost Comparison

Comparing costs after applying discounts and coupons can reveal different savings. There are scenarios when applying a discount first, followed by a coupon, yields a lower price and vice versa. It's all about understanding the math involved.

In the exercise, Mai-Lin uses two methods:

• **Method 1:** Apply the 25% discount first, resulting in $32.49.
• **Method 2:** Apply the $5 coupon first, leading to a final price of $33.74.

By comparing these amounts, it's clear that Method 1—where the discount precedes the coupon—results in a better deal. This occurs because reducing the base price through discounts, and then applying the coupon, leverage more value from the coupon itself.

Therefore, understanding when to apply discounts and coupons optimally ensures the best possible price, saving more money in the long run.