Question

Given a percent of decrease and an initial amount, explain how to find the final amount after the decrease. A large company has 4200 employees. As a result of a declining economy, the company must downsize and let \(6 \%\) of its employees go. How many employees are let go? How many are left?

Short answer

Answer: 252 employees are let go, and 3948 employees remain.

Step-by-step solution

Convert the percent of decrease to a decimal

To convert the percent of decrease into a decimal, divide the percent by 100. In this case, we have a \(6 \%\) decrease in employees, so we divide 6 by 100. \( 6 \% = \frac{6}{100} = 0.06 \)

Find the percentage of employees that remain

To find the percentage of employees that remain, we will subtract the percentage of decrease (in decimal form) from 1. \( 1 - 0.06 = 0.94 \) So, \(94 \%\) of employees will remain.

Calculate the number of employees let go

To find the number of employees who are let go, we will multiply the initial number of employees by the percent of decrease (in decimal form). \( 4200 * 0.06 = 252 \) There are 252 employees who will be let go.

Calculate the final number of employees after the decrease

To find the final number of employees, we will multiply the number of initial employees by the percent of employees that remain (in decimal form). \( 4200 * 0.94 = 3948 \) There will be 3948 employees left after the decrease. Therefore, 252 employees are let go, and 3948 employees remain.

Key concepts

Percentage Conversion

Percentages are a way of expressing a number as a fraction of 100. To work with percentages in most calculations, you first need to convert them into a decimal format. This is because calculations in decimal form are straightforward and easier to handle.
To convert a percentage to a decimal, simply divide the percentage value by 100. For example, if you have a percentage of 6%, you convert it as follows:

• Take the percentage value, which is 6%.
• Divide it by 100: \[\frac{6}{100} = 0.06\].

This process converts the percentage decrease into a decimal form that can then be used in calculations to determine the effect on the initial amount. Doing so allows for calculations like determining how much a certain percentage decrease will affect your initial value. This conversion is an essential step, especially in percentage decrease scenarios where you need to subtract this decimal from a whole to find what remains.

Initial Amount

The initial amount is the starting value or quantity before any changes, such as a decrease or increase, are applied. In problems involving percentages, the initial amount is often the benchmark against which all percentage calculations are made.
In the exercise we are looking at, the initial amount represents the total number of employees before any are let go, which is 4200. This number is crucial because it is multiplied by the percentage of decrease to calculate the number of employees that will be let go due to the downsizing.
Understanding the initial amount helps you gauge the actual impact of a percentage change. It is the reference point for your calculations, determining what you will eventually end up with after adjustments. Always identify your initial value first before applying any percentage changes. This ensures clarity and accuracy in your subsequent calculations.

Final Amount After Decrease

After determining the percentage decrease and understanding the initial amount, the next step is to calculate the final amount remaining. The final amount represents what is left after the specified percentage has been removed from the initial amount.
To find this, you follow these steps:

• First, determine the percentage of the amount that remains after the decrease. This is done by subtracting the decrease from 1. For instance, if the percentage decrease is 6%, the percentage remaining is:\[1 - 0.06 = 0.94\].
• Next, multiply the initial amount by the remaining percentage (in decimal form). Using the example of 4200 employees:\[4200 \times 0.94 = 3948\].

So, the final amount of employees who remain is 3948. Calculating the final amount after a percentage decrease helps provide a clear view of what proportion of the original quantity remains. It combines understanding and application of the initial and converted percentages efficiently.