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Magazine Chapter 6: Problem 1
Rewrite each value as either \(1+r\) or \(1-r\). Then state the rate of increase or decrease as a percent. a. \(1.15\) (a) b. \(1.08\) c. \(0.76\) (i1) d. \(0.998\) e. \(2.5\)
Short Answer
Expert verified
a. 15% increase, b. 8% increase, c. 24% decrease, d. 0.2% decrease, e. 150% increase.
Step by step solution
01
Identifying Increase or Decrease
For each value, determine if it's greater than 1 (indicating an increase) or less than 1 (indicating a decrease). Values equal to 1 indicate no change.
02
Rewrite as 1 卤 r
Rewrite each value in the form of either \(1 + r\) for an increase or \(1 - r\) for a decrease by comparing the value to 1.
03
Convert to Percentage
Express the value of \(r\) found in the previous step as a percentage by multiplying it by 100.
04
Value a. 1.15 Analysis
Since 1.15 is greater than 1, it indicates an increase. Rewrite as \(1 + 0.15\) and convert 0.15 to percentage: 15% increase.
05
Value b. 1.08 Analysis
Since 1.08 is greater than 1, it indicates an increase. Rewrite as \(1 + 0.08\) and convert 0.08 to percentage: 8% increase.
06
Value c. 0.76 Analysis
Since 0.76 is less than 1, it indicates a decrease. Rewrite as \(1 - 0.24\) and convert 0.24 to percentage: 24% decrease.
07
Value d. 0.998 Analysis
Since 0.998 is less than 1, it indicates a decrease. Rewrite as \(1 - 0.002\) and convert 0.002 to percentage: 0.2% decrease.
08
Value e. 2.5 Analysis
Since 2.5 is greater than 1, it indicates an increase. Rewrite as \(1 + 1.5\) and convert 1.5 to percentage: 150% increase.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rate of Change
The concept of rate of change is integral in understanding how quantities vary over time or between conditions. When you hear about percentage increase or decrease, you're actually encountering a form of rate of change. This is crucial in contexts such as finance, science, and even everyday life.
To analyze rate of change:
To analyze rate of change:
- Determine if the quantity is rising or falling. This helps to categorize the change as an increase or decrease.
- Express changes in relation to a base value, often using the form \(1 + r\) for increases and \(1 - r\) for decreases.
- The variable \(r\) here represents the rate of change.
Algebra
In algebra, we often deal with expressions and equations to represent real-world problems in simplified mathematical language. The exercise discussed focuses on transforming numerical values into algebraic expressions of the form \(1 + r\) or \(1 - r\), which simplifies understanding increases and decreases.
Algebra makes these transformations straightforward:
Algebra makes these transformations straightforward:
- Identify the baseline, often set as the number 1, to interpret changes.
- Express relationships using mathematical symbols, such as \(+\) for increases and \(-\) for decreases.
- Solve for \(r\) to find the rate or magnitude of these changes.
Mathematical Concepts
Understanding mathematical concepts is key to mastering problems that involve percentage increases and decreases. These concepts provide the foundational knowledge needed to navigate higher-level math and practical applications.
Here鈥檚 a breakdown:
Here鈥檚 a breakdown:
- Expressions and Equations: These are used to describe and solve problems. By expressing numbers in relation to 1, we create a standardized method to interpret percentage changes.
- Percentages: Converting \(r\) to a percentage involves multiplying by 100. This percentage provides a clear view of how much a value changes relative to its original value.
- Standard Form (\(1 \pm r\)): This form helps identify and clearly state increases and decreases. It simplifies comparisons and calculations, making it easier to convey information.
