Problem 51 - Algebra A Combined Function

Question

A company recently downsized its number of employees by \(35 \%\). If there are still 78 employees, how many employees were there prior to the layoffs?

Step-by-Step Solution

Answer

The company originally had 120 employees.

Step 1: Understand the percentage calculation

When a company downsizes by 35%, it means that 65% of the original number of employees remains (100% - 35%). We know that there are 78 employees remaining, which represents this 65%.

Step 2: Set up the equation

Let \( x \) be the original number of employees. According to the information given, 65% of the original employees still work there. Therefore, we set up the equation: \( 0.65x = 78 \).

Step 3: Solve the equation

Now, we need to solve for \( x \). We do this by dividing both sides of the equation by 0.65: \[ x = \frac{78}{0.65} \]

Step 4: Calculate the result

Perform the division: \[ x = \frac{78}{0.65} \approx 120 \]This means that there were originally 120 employees.

Key Concepts

Solving Equations

Equations are like puzzles. They help us find missing information. In this example, we wanted to figure out the original number of employees before a layoff. We started with an equation. Our unknown value was labeled as \( x \). This represents the original number of employees.
We knew that 65% of the employees remained. Therefore, we created the equation:

• \( 0.65x = 78 \)

Here, 0.65 represents 65% in decimal form. And 78 is the quantity of employees still there after the percentage reduction.
To solve for \( x \), we simply needed to perform division. Divide both sides by 0.65:

• \( x = \frac{78}{0.65} \)

This strategy allows us to isolate \( x \) and find the original number. Once solved, we find \( x = 120 \), indicating that initially, there were 120 employees.

Percentage Reduction

Understanding percentage reduction is crucial in situations like this. When a company downsizes its employees by a certain percentage, it means the workforce decreases by that amount. For example, a 35% reduction means 35% of the original employees are laid off.
To find out how many remain, subtract the reduction percentage from 100%. As shown:

• If a company is reducing by 35%, simply calculate \( 100\% - 35\% = 65\% \).

This tells us that 65% of the original employees are retained. Percentage reductions are common in business. They provide a clear picture of changes in size or quantity. Understanding this helps solve complex problems easily and effectively by converting percentages into manageable numbers for equations.

Employee Downsizing

Employee downsizing refers to reducing the number of employees in a company. Often done to cut costs or improve efficiency. While challenging, it is a common practice in many industries.
To manage downsizing, businesses calculate what percentage of employees need to be released. From the example, the company saw a reduction by 35%. This decision is typically strategic, aiming to retain most essential employees.
Understanding these calculations helps forecast the remaining workforce. After applying the percentage reduction, companies know how many employees will stay.
Downsizing can affect morale and productivity, and it's important for businesses to handle it with transparency. Providing clear communication can help employees understand and adapt to changes. In conclusion, understanding the mathematical aspect of downsizing can aid in planning and decision-making.