Problem 117 - Elementary Algebra

Question

Hernando's salary was \(\$ 49,500\) last year. This year his salary was cut to \(\$ 44,055\). Find the percent decrease.

Step-by-Step Solution

Answer

The percent decrease in Hernando's salary is approximately 11%.

Step 1 - Identify the Initial and Final Values

Determine Hernando's initial and final salaries. Initial salary: \( \$ 49,500 \), Final salary: \( \$ 44,055 \).

Step 2 - Calculate the Decrease in Salary

Subtract the final salary from the initial salary to find the decrease. \( \$ 49,500 - \$ 44,055 = \$ 5,445 \).

Step 3 - Calculate the Percent Decrease

Use the formula for percent decrease: \[ \text{Percent Decrease} = \left( \frac{\text{Decrease}}{\text{Initial Value}} \right) \times 100 \]. Substitute the values: \[ \text{Percent Decrease} = \left( \frac{5,445}{49,500} \right) \times 100 \approx 11\text{%} \].

Key Concepts

initial value

The initial value is an essential concept when calculating percent changes.
It represents the starting point or the original amount before any changes occur.
In this exercise, Hernando’s initial salary of \(\$ 49,500\) is the initial value.
When you understand your initial value, comparing it to later values helps you see how much has changed.

final value

The final value is the amount after a change has occurred.
This value is compared to the initial value to determine the extent of the change.
For Hernando, the final salary after his pay cut is \(\$ 44,055\).
By comparing the final value to the initial value, we can measure the overall change in Hernando’s salary.

percent decrease formula

The percent decrease formula helps us understand the magnitude of a reduction as a percentage of the initial value.
The formula is: \[ \text{Percent Decrease} = \left( \frac{\text{Decrease}}{\text{Initial Value}} \right) \times 100 \]
This formula converts the absolute change into a relative change, expressed as a percentage.
Using Hernando’s salary example, the decrease amounts to \(\$ 5,445\). Using the formula:
\[ \text{Percent Decrease} = \left( \frac{5,445}{49,500} \right) \times 100 \approx 11\% \]
This tells us that Hernando's salary was decreased by 11% from the original amount.

subtraction in algebra

Subtraction in algebra helps us find the difference between two numbers or values.
It is a basic yet crucial operation for solving many problems, including percent changes.
In this problem, we subtract the final salary from the initial salary to find the decrease: \(\$ 49,500 - \$ 44,055 = \$ 5,445\).
This difference, called the decrease, is then used to calculate the percent decrease.
Understanding how to perform subtraction accurately is fundamental to solving such mathematical problems.