Question

During a sale, a store decreases the prices on all of its scarves by 25 to 50 percent. If all of the scarves in the store were originally priced at \(\$ 20,\) which of the following prices could be the sale price of a scarf? Indicate all such prices. a. \(\$ 8\) b. \(\$ 10\) c. \(\$ 12\) d. \(\$ 14\) e. \(\$ 16\)

Short answer

The possible sale prices of a scarf could be $10, $12, or $14.

Step-by-step solution

Calculate minimum possible price

The minimum possible price of a scarf can be calculated by reducing the original price by 50%. In calculation terms, it would be \(20 - (50/100 * 20) = 10\). So the lowest possible price of a scarf is $10.

Calculate maximum possible price

The maximum possible price of a scarf can be calculated by reducing the original price by 25%. In calculation terms, it would be \(20 - (25/100 * 20) = 15\). So the highest possible price of a scarf is $15.

Compare the Options with the Calculated Range

Now compare the given price options with the calculated range. Only options falling within this range would be possible sale prices of a scarf. Upon comparison, we find that the options \(10\), \(12\), \(14\) fall within this range.

Key concepts

Understanding Percentage Decrease

In any sale scenario, a percentage decrease is when the original price is reduced by a certain percentage amount. For example, a 25% decrease means you remove 25% of the original price. To find this decrease, we multiply the original price by the percentage in decimal form.
Let's consider the pricing scenario of the scarves that originally cost $20. For a 25% decrease, multiply 20 by 0.25, which is the decimal form of 25%. This gives us a decrease of $5, so the sale price would be $15.
Similarly, a 50% decrease means the price is cut in half, so you multiply 20 by 0.50, obtaining a $10 decrease. Subtract this from the original price of $20 to get the sale price of $10. Understanding how to calculate percentage decreases is crucial for evaluating sale prices.

Calculating the Price Range

When given a range of potential discounts, it’s useful to calculate the lowest and highest possible prices. This allows us to see which prices are feasible within the range of discounts offered.
In the example, to find the lowest price, calculate a 50% decrease from the original price of $20, getting a minimum price of $10. To find the highest price, calculate a 25% decrease, which results in a $15 price.
The valid sale prices must then fall between $10 and $15. This calculation is fundamental when determining feasible prices during sales, ensuring you stay within the proper discount boundaries.

Steps in Mathematical Problem Solving

Solving problems systematically using mathematical reasoning comprises several steps. First, comprehend the problem and identify what is being asked. In this exercise, it's finding possible discounted prices. Next, develop a strategy, such as calculating the percentage decrease to get a range of possible prices.
After calculations, compare the results to see if they align with the given options. This means checking each potential price against the feasible range. Finally, validate your results to ensure they have been computed correctly. This methodical problem-solving approach helps in breaking down complex mathematical problems into manageable steps.

Enhancing Reasoning Skills

Reasoning skills involve the ability to think logically and make connections between different concepts. In the scarf pricing exercise, reasoning is used in several ways.

• Deciphering the intent of the problem, which is to determine valid sale prices.
• Understanding the application of percentage reductions and their calculated results.
• Making informed decisions by comparing the calculated price range with the provided options.

Improving reasoning skills can be achieved by practicing similar exercises, which build confidence in your ability to assess problems and develop appropriate solutions.
These skills are not only invaluable for math but also for any problem-solving scenario in life.