Problem 148 - Elementary Algebra

Question

Stella bought a dinette set on sale for \(\$ 725\). The original price was \(\$ 1,299 .\) To the nearest tenth of a percent, what was the rate of discount?

Step-by-Step Solution

Answer

The rate of discount is 44.2%.

Step 1: Find the amount of discount

The amount of discount can be found by subtracting the sale price from the original price. \[ \text{Discount Amount} = \text{Original Price} - \text{Sale Price} \] \[ \text{Discount Amount} = 1299 - 725 = 574 \]

Step 2: Calculate the rate of discount

To find the rate of discount, divide the discount amount by the original price and then multiply by 100 to convert it to a percentage. \[ \text{Rate of Discount} = \frac{\text{Discount Amount}}{\text{Original Price}} \times 100 \] \[ \text{Rate of Discount} = \frac{574}{1299} \times 100 \]

Step 3: Simplify the division and multiplication

Perform the division and multiplication to find the percentage. \[ \frac{574}{1299} \times 100 \ \frac{574}{1299} \ \text{Rate of Discount} \ \text{Rate of Discount}= 44.2 \]

Key Concepts

discount calculation

Calculating the discount is an essential part of many financial transactions, especially when buying items on sale. The first step is to determine the Discount Amount, which is the difference between the Original Price and the Sale Price. This can be easily obtained by subtracting the two values.

For instance, if an item originally costs \$1299\ and is on sale for \$725\, the Discount Amount would be:
\[ \text{Discount Amount} = 1299 - 725 = 574 \]

Now, we know that Stella saved \$574\ on the purchase.

Understanding how to calculate this value helps in multiple everyday scenarios, from budgeting to shopping more effectively.

percentage conversion

Once we have the Discount Amount, we need to convert it into a percentage to understand the rate of discount better. Percentages are incredibly useful as they provide a sense of proportion or ratio, making numbers easier to compare and understand.

To convert the Discount Amount to a percentage, you'll use the formula:
\[ \text{Rate of Discount} = \frac{\text{Discount Amount}}{\text{Original Price}} \times 100 \]

In the example, with a Discount Amount of \$574\ and an Original Price of \$1299\, the calculation is:
\[ \text{Rate of Discount} = \frac{574}{1299} \times 100 \]

By performing this division and multiplication, you'll find the percentage rate, providing a clear representation of the discount relative to the original cost.

basic algebra

Basic algebra is the foundation of solving various problems, including those involving discounts and financial calculations. Skills in algebra include manipulating equations and understanding arithmetic operations.

In our example, we use algebraic steps to simplify the Rate of Discount formula:
\[ \frac{574}{1299} \times 100 \].

Performing the division \( \frac{574}{1299} \) gives approximately \ 0.442 \. When we multiply this by 100, we get: \[ 0.442 \times 100 = 44.2 \]

Hence, the rate of discount is \ 44.2\% \.

Mastering these basic algebraic techniques is essential not only for schoolwork but also in practical everyday activities, such as calculating discounts, interest rates, and other financial figures. Basic algebra simplifies otherwise complex problems into manageable steps.