Question

Lonnie announces a sale of \(\frac{1}{3}\) off all merchandise and puts special sale tags on every item. Toward the end of the sale, she decides to give customers an additional \(\frac{1}{3}\) off the marked-down prices. What is the total discount? A. \(\frac{1}{9}\) of the original price B. \(\frac{2}{3}\) of the original price C. \(\frac{5}{9}\) of the original price D. \(\frac{4}{9}\) of the original price

Short answer

C. \(\frac{5}{9}\) of the original price.

Step-by-step solution

Initial Markdown Calculation

The initial discount offered is \(\frac{1}{3}\) off the original price. Let the original price be represented as \( P \). Therefore, the price after the first discount is: \[ P - \frac{1}{3}P = \frac{2}{3}P \].

Additional Markdown Calculation

The additional discount is also \(\frac{1}{3}\) off the already marked-down price. So, the price after the additional discount is: \[ \frac{2}{3}P - \frac{1}{3}(\frac{2}{3}P) = \frac{2}{3}P - \frac{2}{9}P \].

Simplify the Final Expression

Simplify the expression from the previous step to determine the final price. Combine the fractions: \[ \frac{2}{3}P - \frac{2}{9}P = \frac{6}{9}P - \frac{2}{9}P = \frac{4}{9}P \].

Calculate Total Discount

The final price is \(\frac{4}{9}\) of the original price. Therefore, the total discount is: \[ P - \frac{4}{9}P = \frac{5}{9}P \].

Conclusion

Thus, the total discount off the original price is \(\frac{5}{9}\).

Key concepts

fractions

Fractions represent a part of a whole. They are written as two numbers separated by a line, with the top number called the numerator and the bottom number called the denominator. In our exercise, we deal with fractions like \(\frac{1}{3}\), representing one part out of three. Simplifying fractions can involve finding a common denominator and combining values.

percentage discount

Percentage discounts are common in sales and represent a reduction relative to the original price. Here, we converted the percentage to a fraction for easier calculation. Initially, we had a \(\frac{1}{3}\) discount off the original price, leaving us with \(\frac{2}{3}\) of the price. Then, another \(\frac{1}{3}\) discount was applied to the marked-down price. Finally, we calculated the cumulative effect of these combined discounts.

step-by-step solutions

Breaking down problems into smaller steps helps in understanding and solving them. Start with determining what each part of the question means. In our example:
1. Calculate the initial discount.
2. Apply the additional discount.
3. Simplify the expressions. By following a structured approach, it clarifies the path from problem to solution and makes complex calculations easier to handle.

mathematical reasoning

Mathematical reasoning involves logical thought processes to solve problems. It can include recognizing patterns, making deductions, and simplifying expressions. With our exercise, we reason through each discount step-by-step, combine and simplify fractions to find the net effect of combined discounts. This approach can be applied to many types of problems, ensuring consistent and reliable solutions.