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 prices for a scarf on sale are \$10 and \$12, so the correct answers are options B and C.

Step-by-step solution

Identify the Original Price

The original price of each scarf is \$20.

Compute the Prices After the Maximum and Minimum Discounts

First, calculate the price after a 25 percent discount, and then after a 50 percent discount. A 25 percent discount on \$20 is \$5 (\(0.25 \times 20\)), so the price of the scarf after this discount is \$15, i.e., \(20 - 5\). A 50 percent discount on \$20 is \$10 (\(0.50 \times 20\)), so the price of the scarf after this discount is just \$10, i.e., \(20 - 10\).

Compare Given Prices With Range

Now compare the prices given in the options with the price range calculated in step 2. Prices that are within the range from \$10 to \$15, inclusive, are the possible sale prices of a scarf.

Key concepts

Percentage Discounts

When you see a price tag that shows a percentage discount, it essentially tells you how much of the item's original price you are saving. Let's break down how this works in simple terms. Suppose an item costs $100, and there's a 20% discount. Here's how you calculate that discount:

• First, convert the percentage into a decimal. (20% becomes 0.20)
• Next, multiply the original price by this decimal: 100 x 0.20 = $20
• This means you save $20, and you pay: 100 - 20 = $80

Understanding how to apply percentages is crucial for figuring out how much you're actually paying after a discount. In sales, discounts are often between 0% and 100%. The higher the percentage, the more you save!

Price Calculations

Calculating the final price after a discount involves a few straightforward steps. Let's apply these to the concept of scarves on sale. Assume that every scarf costs $20 originally. If there's a 25% discount, the first step is finding 25% of 20:

• Convert 25% into decimal form: 0.25
• Multiply the original price: 20 x 0.25 = $5

Thus, the discount is $5, and the new price of the scarf becomes: 20 - 5 = $15 For a 50% discount, repeat the steps:

• Convert 50% into decimal: 0.50
• Multiply the original price: 20 x 0.50 = $10

Therefore, the price after a 50% discount is: 20 - 10 = $10 By these calculations, any price within this discounted range for our scarves would fit under the parameters provided.

Problem-solving Steps

Approaching math problems systematically can make them less intimidating. Here's a simple way to tackle problems related to discounts and pricing. 1. **Identify the Original Price:** - Know how much the item costs before any discounts. For example, the original price of each scarf is $20. 2. **Calculate the Discounted Prices:** - Determine the range of possible prices after applying discounts (from 25% to 50% in our case). - Perform separate calculations for the minimum and maximum discounts to define this range. 3. **Match with Possible Prices:** - Compare these calculated prices to other given options. This defines your possible prices during sales. Following these problem-solving steps will guide you in understanding and solving similar problems efficiently.

Price Range Determination

Price range determination helps in understanding the limits within which a sale price might fall. This is how you can determine it step-by-step: First, calculate the price after the minimum discount (25% for our scarves). We found this to be $15. Then, calculate the price after the maximum discount (50%), which was $10. These calculations create the price range. So, during the sale, these scarves can cost anywhere from $10 to $15. In exercises, when asked to find possible sale prices, look for options falling within this range. These are the prices that align with the discount range. So for the exercise above, the viable sale prices would be $10 and $12, falling within $10-$15.