Problem 14 - Prealgebra

Question

Find the percent of change. Round to the nearest tenth, if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. from \(257 \mathrm{m}\) to \(243 \mathrm{m}\)

Step-by-Step Solution

Answer

The percent change is a 5.4% decrease.

Step 1: Identify the Original and New Values

The original value is the initial measurement, which is 257 meters. The new value is the measurement it changes to, which is 243 meters.

Step 2: Calculate the Amount of Change

Find the difference between the original value and the new value. This is calculated by subtracting the new value from the original value: \[\text{Amount of Change} = 257 - 243 = 14\]

Step 3: Find the Percent of Change

To find the percent of change, divide the amount of change by the original value and then multiply by 100 to convert it into a percentage:\[\text{Percent of Change} = \left( \frac{14}{257} \right) \times 100 \approx 5.4\]

Step 4: Determine If It's an Increase or Decrease

Since the original value (257) is greater than the new value (243), this is a percent of decrease.

Key Concepts

Understanding Percent Increase

In mathematics, a percent increase occurs when a value rises above its original amount. It helps to understand how much growth has happened in relation to the starting figure. Here's how you spot and calculate a percent increase:

• Subtract the original value from the new value to find the increase amount.
• Divide the increase amount by the original value. This gives you the decimal form.
• Convert the decimal to a percentage by multiplying it by 100.

For instance, if something grows from 100 to 120, the increase is 20. This would be a 20% increase, as \(\left( \frac{20}{100} \right) \times 100 = 20\). A handy tip is to check if the original value is smaller than the new value; then it's a likely increase.

Understanding Percent Decrease

A percent decrease happens when a value falls from its original amount. It gives you a sense of how much reduction occurs compared to the initial figure. Calculating a percent decrease involves a few simple steps:

• Subtract the new value from the original value to find the decrease amount.
• Divide this decrease amount by the original value to get the decimal form.
• Multiply the result by 100 to convert it into a percentage.

In our example of 257 meters decreasing to 243 meters, there's a change of 14 meters, amounting to a 5.4% decrease, calculated as \(\left( \frac{14}{257} \right) \times 100 \approx 5.4\). Remember, if the original value is larger than the new value, it's a clear indication of a decrease.

The Role of Mathematics in Understanding Changes

Mathematics is pivotal when it comes to understanding real-world changes in values, such as measuring percentage changes. It helps break down complex ideas into simple computations that reveal meaningful insights. Whether it's a price drop or an increase in population, math gives us the tools to measure and make sense of these changes. Knowing how to manipulate numbers through basic operations like subtraction, division, and multiplication allows us to comprehend changes effectively. Mathematics equips students and professionals alike to analyze and interpret data, making it an indispensable skill in everyday life. By learning to calculate percent changes, you develop the ability to identify trends and make informed decisions based on those trends.

Problem Solving Through Percentage Calculations

Calculating percent changes is a powerful problem-solving skill. It simplifies identifying trends and patterns, which can guide decisions in finance, science, and general day-to-day decisions. For students, the practice becomes a fantastic way to hone their analytical skills. Here are some quick tips:

• Always identify the original and new values accurately.
• Don't forget to multiply by 100 to express a change as a percentage.
• Double-check whether you're dealing with an increase or decrease, as this affects interpretation.

Engaging with percent calculations pushes us to think critically and apply basic math to solve real-world problems. By mastering percent change calculations, students enhance not only their numerical literacy but also their problem-solving capabilities, leading to better reasoning and analytical skills.