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Chapter 6: Problem 44

On each of the 22 work days in a particular month, every employee of a start- up venture was sent a company communication. If a total of 4642 total company communications were sent, how many employees does the company have, assuming that no staffing changes were made that month?

Short Answer

Expert verified
The company has 211 employees.

Step by step solution

01

Identify the variables

Let the number of employees in the company be denoted by the variable \( E \). The problem states that each employee receives one communication per day for 22 days.
02

Set up the equation

Because each employee receives a communication each day, the total number of communications sent in 22 days can be represented as: \( 22 \times E \).
03

Write the total communications expression

The total number of communications sent is given as 4642. Therefore, we have the equation: \( 22E = 4642 \).
04

Solve for \( E \)

To find \( E \), divide both sides of the equation by 22: \( E = \frac{4642}{22} \).
05

Calculate the result

Perform the division to find \( E \): \( E = 211 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Setup
Setting up an equation is often the first and most crucial step in solving a mathematical problem. In this exercise, we start with understanding the relationship given by the problem. Each employee receives one communication per day for 22 days. To set up our equation:
  • We multiply the number of employees (denoted as \(E\)) by the number of days (22).
  • This gives us the expression for the total number of communications sent.
This approach translates the problem's conditions into a numerical expression which we can solve.
Variable Identification
Identifying variables is critical to simplify and solve equations. In this context:
  • We let the number of employees be represented by the variable \(E\).
  • This helps us keep track of what we are solving for and makes the equation easier to manipulate.
By isolating \(E\), we ensure our calculations target the number of employees.
Variables act as placeholders that can simplify expressions and make it clear what each part of the equation represents.
Linear Equations
Linear equations are equations of the first order and their graph is always a straight line. In this exercise,\( 22E = 4642\) is a linear equation. Here's why:
  • The equation represents a simple relationship between two variables with a constant coefficient (22).
  • This kind of equation can be solved by basic arithmetic operations (addition, subtraction, multiplication, and division).
Linear equations are foundational in mathematics because they cover constant rates of change, making them easier to solve and understand.
Basic Arithmetic Operations
Arithmetic operations are the building blocks of most mathematical problems. To solve our linear equation, we need to perform the following steps:
  • First, we isolate the variable \(E\) by dividing both sides of the equation by 22: \(E = \frac{4642}{22}\).
  • Next, we perform the division: \(E = 211\).
These steps fall under basic arithmetic operations: multiplication and division. They are straightforward but crucial for solving for the variable correctly.
Always ensure you perform these operations carefully to avoid errors in your final answer.

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