Problem 21 - Prealgebra

Question

OPEN ENDED Give an example of a percent of decrease.

Step-by-Step Solution

Answer

The percent of decrease is 30%.

Step 1: Identify Original and New Values

Let's say that a sweater was originally priced at $50, but it is now on sale for $35. Here, the original value is $50, and the new value is $35.

Step 2: Calculate the Absolute Decrease

Subtract the new value from the original value to find the absolute decrease. In this case, subtract \(35 from \)50: \[ 50 - 35 = 15 \] Thus, the absolute decrease is $15.

Step 3: Calculate the Percent Decrease

To find the percent decrease, divide the absolute decrease by the original value, and multiply by 100 to convert it to a percentage. \[ \left( \frac{15}{50} \right) \times 100 = 30\% \] Therefore, the percent of decrease is 30%.

Key Concepts

Understanding Percent Calculations

Percent calculations are an essential tool in mathematics that help us express numbers as a part of a whole. To find a percent value, you generally divide a part by the whole and multiply by 100. For example, in the case of a price decrease, you divide the decrease amount by the original price, and then multiply by 100 to find the percentage.

This process is used not only in discounts but also in situations like population change, test score improvements, and more. Understanding how to convert between fractions, decimals, and percentages is crucial.

• To convert a fraction to a percentage, multiply by 100.
• To convert a decimal to a percentage, also multiply by 100.
• Percent indicates "per hundred," and calculations always relate it back to 100.

Using a systematic two-step process—finding the absolute change and then calculating the percentage change—can simplify resolving such tasks.

Grasping Absolute Decrease

The absolute decrease refers to the raw difference between two numeric values. It represents the direct subtraction of the smaller number from the larger number without any additional calculations.

In our example, the original price of the sweater was $50, and the sale price is $35. The absolute decrease is straightforward: subtract $35 from $50 to get $15.

• It tells you exactly how much a value has decreased.
• No percentages or proportions are involved at this stage.
• The focus is on the actual amount decreased.

The absolute decrease is a necessary step before you can find the proportional decrease, as it gives you the "change" number necessary for the percentage calculation.

The Role of Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to understand and solve problems. It includes forming strategies to approach and simplify problems, like calculating a percent decrease. This reasoning involves both critical thinking and the systematic approach commonly used in mathematical problems.

In the context of percentage decrease, we use reasoning to logically deduce how changes in one quantity relate to the original. This understanding helps build a coherent step-by-step process: identify the baseline value, find the change, and express that change relative to the original.

• Reasoning involves identifying clear logical steps.
• Helps simplify complex word problems.
• Promotes understanding over memorization.

Through mathematical reasoning, not only do you arrive at the correct numerical answer, but you also gain a deeper understanding of the concept in real-world scenarios.