Problem 76 - Algebra 2

Question

Find each percent of increase or decrease. from 9 \(\mathrm{m}\) to 10 \(\mathrm{m}\)

Step-by-Step Solution

Answer

The percent increase from 9 m to 10 m is 11.11%.

Step 1: Calculate the Change in Value

Find the difference between the two measurements. Subtract the original value from the new value. Change in value = New Value - Original Value = 10 m - 9 m.

Step 2: Calculate the Percent Change

Use the formula for percent change: Percent Change = (Change in Value / Original Value) * 100%. Plug in the numbers: (1 m / 9 m) * 100%.

Step 3: Perform the Calculation

Calculate the percent change. (1/9) * 100% = 11.11% (rounded to two decimal places).

Key Concepts

Percent Change Calculation

Understanding how to calculate the percent of increase or decrease between two values is a fundamental concept in various fields, including mathematics, economics, and business. In the given exercise, the aim is to discern how much an initial measurement has grown or reduced in comparison to a new measurement.

To achieve this, we first determine the absolute change by subtracting the original value from the new one. For example, to find out the change when length increases from 9 meters to 10 meters, subtract 9 m from 10 m, giving us a change of 1 meter. This step establishes the basis for the subsequent percent change calculation, which will illustrate the proportional difference relative to the original amount.

It is important to note that the calculation of percent decrease follows the same formula as that for percent increase. The only difference is that the change in value would be negative when there's a decrease, thus leading to a negative percent change. Clarifying this distinction helps students understand that the foundation for calculating both increase and decrease is the same algebraic operation.

Percent Change Formula

The percent change formula is a mathematical expression used to calculate the percentage difference between an old and a new value. The formula is expressed as: \
Percent Change = (Change in Value / Original Value) \times 100%
\
To apply this formula, divide the change in value by the original value. Multiplying the result by 100 converts it from a decimal to a percentage, which is easier to interpret. For instance, in our example, the change in value is 1 m and the original value is 9 m. Inserting these into the formula gets us \( \frac{1 m}{9 m} \times 100% \) which simplifies to approximately 11.11% when rounded to two decimal places.

This formula is versatile, applying to any scenario where you need to express change as a percentage. As a tip for students, it is crucial to keep the value in the same unit or to have them converted appropriately before using the formula, to ensure the accuracy of the calculation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent mathematical relationships. These expressions are the backbone of algebra and vital for solving problems that involve variables.

In the context of calculating percent change, the algebraic expression encapsulates the relationship between the original value, the new value, and the change in value. For instance, the expression \( (New Value - Original Value) / Original Value \times 100% \) summarizes the process for finding percent change. Each part of this expression plays a specific role in the calculation: subtraction determines the change in value, division computes the magnitude of this change relative to the original value, and multiplication by 100 converts the fraction into a percentage.

By becoming familiar with algebraic expressions, students can understand the underlying structure of the problem, and more importantly, apply this understanding to a wide array of situations where calculation of percent change is needed. Developing proficiency in interpreting and manipulating these expressions is a valuable skill that extends beyond mathematics into daily practicalities such as calculating discounts, interest rates, or growth rates.