Problem 3 - Intermediate Algebra

Question

Fill in the blank: If a job can be completed in \(x\) hours, then the rate of work can be expressed as \(\frac{1}{\underline{\phantom{xx}}}\) of the job is completed per hour.

Step-by-Step Solution

Answer

The rate of work can be expressed as \(\frac{1}{x}\) of the job per hour.

Step 1: Understanding the Problem

The problem asks us to find how we can express the rate of work done if a job takes \(x\) hours to complete. The hint given is that the rate of work can be expressed as a fraction of the job completed per hour.

Step 2: Concept of Rate of Work

The rate of work is generally expressed as the portion of the task completed in a unit of time. If a task takes \(x\) total hours to complete, then in 1 hour, a fraction of \(\frac{1}{x}\) of the task is completed.

Step 3: Filling the Blank

To complete the statement with the correct expression, we observe that each hour, \(\frac{1}{x}\) of the job is done. Therefore, the blank should be filled with \(x\), resulting in the expression \(\frac{1}{x}\) of the job is completed per hour.

Key Concepts

Understanding Time and Work Problems

Time and work problems are a fascinating and practical math concept that illustrate how time and effort are distributed in finishing a task. These types of problems evaluate how long it takes for one or more workers to complete a job together or individually. They involve understanding the relationship between time, work, and the rate at which work is completed.

To solve such problems, the total work needed to finish a job is often divided into parts, and the amount each worker can complete in a certain time is evaluated. It is important to break down the job into smaller, more manageable units when calculating individual contributions per unit of time.

Consider a scenario where a gardener can plant a garden in 5 hours. In time and work problems, it becomes relevant to ask: **How much of the garden can the gardener plant in 1 hour?** Solving problems like these help us see how to distribute workloads and optimize efficiency.

Exploring the Work Rate Formula

The work rate formula is a vital tool in solving time and work problems. This formula establishes a clear connection between the total amount of work, the time taken, and the rate of work completion. The basic idea is that *rate of work* is merely the fraction of work done per time unit.

In mathematical terms, if a job requires \( x \) hours to complete, the work rate formula tells us that the rate of work is given by \( \frac{1}{x} \) of the job per hour. This formula highlights how much of the job is completed for each hour worked.

**Key aspects of the formula include:**

• The numerator (1) represents a single unit of work done, usually considered as the entire task.
• The denominator (x) represents the total time in hours required to complete the task.
• The fraction \( \frac{1}{x} \) is the rate representing portions of the work finished in one hour.

In essence, this formula allows one to compare different rates and estimate significant factors like duration and productivity.

Calculating the Fraction of Job Completed

When working with time and work problems, determining the fraction of the job completed at any given time is crucial. This involves understanding how the work rate contributes to the gradual progression of task completion.

Suppose you have a task that takes \( x \) hours to finish. After each hour, \( \frac{1}{x} \) of the task is completed. This means after 2 hours, the fraction of the job completed would be \( 2 \times \frac{1}{x} = \frac{2}{x} \). This pattern continues, scaling up linearly with time, making it simple to determine how much work has been done at any point.

Here’s how you can apply this:

• To find the fraction of the work complete after 4 hours when \( x = 8 \), you calculate \( \frac{4}{8} = \frac{1}{2} \). So, half the task is done.
• Following this fractional approach helps in monitoring task progress and estimating remaining effort efficiently, which is handy for project planning and management.

This concept of fractional completion helps in clearly visualizing the parts of the work finished over time, essential for both individual and collaborative tasks.