Problem 86 - Elementary Algebra

Question

Problem: Complete the "Understand the problem," "Make a plan," and "Complete the plan" steps. The regular price of a 12 -cup programmable coffee maker is \(\$ 79.99\). The sale price is \(\$ 59.99\). Find the percent discount. Round to the nearest percent. Incorrect Answer: The unknown is the percent discount; \(P=\) percent discount. A word equation is percent decrease \(=\left(\frac{\text { decrease in price }}{\text { sale price }}\right)(100) \%\). $$ \begin{aligned} &P=\left(\frac{\$ 79.99-\$ 59.99}{\$ 59.99}\right)(100) \% \\ &P=33 \% \text { decrease } \end{aligned} $$

Step-by-Step Solution

Answer

The percent discount is 25%.

Step 1 - Understand the Problem

The goal is to find the percent discount on a coffee maker. The original price is \( \$79.99 \) and the sale price is \( \$59.99 \).

Step 2 - Identify the Decrease in Price

Calculate the decrease in price by subtracting the sale price from the original price. \[ \text{Decrease in price} = \$79.99 - \$59.99 = \$20.00 \]

Step 3 - Set up the Percent Discount Formula

The percent discount can be found using the formula: \[ \text{Percent Discount} = \left( \frac{\text{Decrease in Price}}{\text{Original Price}} \right) \times 100 \]

Step 4 - Substitute Numbers into the Formula

Plug in the values into the formula: \[ \text{Percent Discount} = \left( \frac{\$20.00}{\$79.99} \right) \times 100 \]

Step 5 - Simplify the Expression

Simplify the fraction and multiply by 100 to find the percent discount: \[ \text{Percent Discount} \approx 0.25 \times 100 = 25 \% \]

Step 6 - Round to the Nearest Percent

Since the decimal is less than 0.5, round down to get the final answer: 25%.

Key Concepts

percent decrease

Understanding how to calculate a percent decrease is very useful. When we talk about percent decrease, we are referring to how much something has reduced in terms of a percent.
For example, if the price of a coffee maker drops from \$79.99\ to \$59.99\, we want to find out what percentage of the original price this decrease represents.
The first step in finding the percent decrease is to identify the decrease in amount. For our example:
\[ \text{Decrease in Price} = \$79.99 - \$59.99 = \$20.00 \ \].
Next, we find what fraction of the original price this decrease represents: \[ \frac{\text{Decrease}}{\text{Original Price}} = \frac{\$20.00}{\$79.99} \ \].
Finally, to convert this fraction to a percentage, we multiply by 100:
\[ \text{Percent Decrease} = \frac{\text{Decrease}}{\text{Original Price}} \times 100 \].

discount formula

Using the correct discount formula can make calculating percent discounts much simpler. Look at the formula closely: \[ \text{Percent Discount} = \frac{\text{Decrease in Price}}{\text{Original Price}} \times 100 \ \].
This formula tells us that to find the percent discount, we need to:

• Calculate the amount of the decrease, which is the original price minus the sale price.
• Divide the decrease by the original price to get a fraction.
• Multiply the fraction by 100 to convert it to a percent.

As seen in our example:

• Original Price: \$79.99\
• Sale Price: \$59.99\
• Decrease: \$20.00\

Substitute these values in the formula: \[ \text{Percent Discount} = \frac{\$20.00}{\$79.99} \times 100 = 25\begin{ aligned } \right) \].

rounding percents

Rounding percents helps to simplify the answer, making it more understandable. When we calculate the percent of something, the result can be a long decimal number. To make it easier to communicate, it's common to round this number to the nearest whole percent. Here's a simple rule for rounding:

• If the decimal part is 0.5 or greater, round up.
• If it's less than 0.5, round down.

In our coffee maker example, we calculated the percent discount to be approximately 25.00%. Since the decimal is not more than 0.5, we round down, making the answer 25%. This means the coffee maker is sold at 25% off the original price.