Question

Elita bought a scarf on sale for \(\$ 14.49\) after a \(16 \%\) discount. What was the scarf's original price? A. \(\$ 17.25\) B. \(\$ 17.34\) C. \(\$ 12.88\) D. \(\$ 14.95\) The following question contains a blank marked \begin{tabular}{|l|l|} \hline Select... & . Beneath the blank is \(a\) set of choices. \\ \hline \end{tabular} Indicate the choice that is correct and belongs in the blank. (Note: On the real GED* test, the choices will appear as \(a\) "drop-down"menu. When you click on a choice, it will appear in the blank.)

Short answer

The original price was \$ 17.25 (option A).

Step-by-step solution

Understand the Discount Formula

The formula for the discounted price is given by: \[\text{Discounted Price} = \text{Original Price} \times (1 - \text{Discount Rate})\]. In this problem, the discounted price is \$ 14.49 and the discount rate is 16%.

Setup the Equation

Let the original price be represented by \( P \). Set up the equation based on the discount formula: \[14.49 = P \times (1 - 0.16)\].

Simplify the Equation

Simplify the equation using the discount rate: \[14.49 = P \times 0.84\].

Solve for the Original Price

To find \( P \), divide both sides of the equation by 0.84: \[P = \frac{14.49}{0.84}\]. Using a calculator, we find: \[P = 17.25\].

Validate the Original Price

Verify that \( P = 17.25 \) fits the given choices. The correct answer is \( \$ 17.25 \).

Key concepts

Discount Formula

Understanding the discount formula is crucial. It helps you determine how much you save when an item is on sale. The basic discount formula is:
\[ \text{Discounted Price} = \text{Original Price} \times (1 - \text{Discount Rate}) \]
Here, the Original Price is the initial cost of the item before any reductions. The Discount Rate is the percentage taken off. This formula multiplies the original price by a factor that subtracts the discount rate to find the new price.
For example, if an item costs \(50 and has a 20% discount:
\[ \text{Discounted Price} = 50 \times (1 - 0.20) = 50 \times 0.80 = 40 \]
Thus, you pay \)40 for the item after the discount.

Solving Equations

Solving equations involves finding a value for the variable that makes the equation true. In this problem, we need to find the original price of the scarf. Let's break it down step-by-step.
1. **Set Up the Equation:** Let the original price be represented by \( P \). Our equation based on the discount formula is:
\[ 14.49 = P \times (1 - 0.16) \]
2. **Simplify the Equation:** Calculate the value inside the parentheses:
\[ 14.49 = P \times 0.84 \]
3. **Solve for the Variable:** To isolate \( P \), divide both sides by 0.84:
\[ P = \frac{14.49}{0.84} \]
Using a calculator:
\[ P = 17.25 \]
This means the original price of the scarf was $17.25 before the discount.
4. **Validate Your Solution:** Always check your answer. A quick way is to multiply 17.25 by 0.84 to see if it equals 14.49. This confirms our solution is correct.

Percentages in Math

Percentages play a crucial role in real-life scenarios like discounts, interest rates, and more. To understand them clearly:
- **Percentage** means per hundred. So, 16% is the same as 16 out of 100, or 0.16.
- **Converting Percentages:** To use percentages in calculations, convert them to decimals. For 16%, divide by 100:
\[ 16\text{\text{ percent}} = \frac{16}{100} = 0.16 \]
- **Using Percentages:** When you see a discount of 16% off, it tells you to pay only 84% of the original price. Hence, we use the equation:
\[ \text{Discounted Price} = \text{Original Price} \times (1 - 0.16) = \text{Original Price} \times 0.84 \]
Understanding these principles helps you navigate many financial decisions more effectively.