Problem 52 - Algebra A Combined Function

Question

One person proofreads copy for a small newspaper in 4 hours. If a second proofreader is also employed, the job can be done in \(2 \frac{1}{2}\) hours. How long does it take for the second proofreader to do the same job alone?

Step-by-Step Solution

Answer

The second proofreader can complete the job alone in 6 hours and 40 minutes.

Step 1: Determine the Rate of the First Proofreader

A person's rate is typically calculated as the Reciprocal of their time on a job. Since the first proofreader can complete the job in 4 hours, their rate is \(\frac{1}{4}\) of the job per hour.

Step 2: Determine the Rate of Both Proofreaders Together

The combined rate of both proofreaders is also the reciprocal of the time it takes them when working together. Since they can complete the job in \(2 \frac{1}{2}\) hours when both working together, the combined rate is \(\frac{1}{2.5} = \frac{2}{5}\) of the job per hour.

Step 3: Establish an Equation for the Second Proofreader's Rate

Let \(r\) represent the rate of the second proofreader. Together, the guidance from the rates combined is: \(\frac{1}{4} + r = \frac{2}{5}\).

Step 4: Solve for the Second Proofreader's Rate

Rearrange \(\frac{1}{4} + r = \frac{2}{5}\) to isolate \(r\). \[ r = \frac{2}{5} - \frac{1}{4} \]Calculate the values: \(\frac{2}{5} = \frac{8}{20}\) and \(\frac{1}{4} = \frac{5}{20}\). Thus, \[ r = \frac{8}{20} - \frac{5}{20} = \frac{3}{20} \].So, the rate of the second proofreader is \(\frac{3}{20}\) of the job per hour.

Step 5: Calculate the Time for the Second Proofreader Alone

The time required for the second proofreader to complete the entire job when working alone is the reciprocal of their rate. This is \(\frac{1}{r} = \frac{1}{\frac{3}{20}} = \frac{20}{3}\) hours, which simplifies to \(6\) hours and \(40\) minutes.

Key Concepts

Work Rate

A work rate essentially refers to how much of a job one can complete in a given time frame. In the context of proofreading, the work rate is often expressed in terms of jobs per hour. It's a crucial concept in rate problems because it helps determine how long it takes to complete a task alone or in a team.

For example, if a proofreader can complete a task in 4 hours, we calculate their work rate as the reciprocal of the time taken, which is one-fourth \( \frac{1}{4} \) of the job per hour. Understanding this allows us to establish a baseline for how much of the work is done by one person and paves the way to calculate contributions from additional people.

Combined Work Rate

When two or more people work together on a task, their combined efforts speed up the task completion time. The combined work rate is calculated by adding the individual rates of all involved.

In our proofreading example, when two proofreaders work together and finish the task in \(2 \frac{1}{2}\) hours, their combined rate is \( \frac{2}{5} \) of the task per hour. This step is crucial because it reflects how pooling resources can achieve more than individual efforts alone.

Algebraic Equations

Algebraic equations are employed to represent relationships between different elements in work rate problems. These equations provide a structured approach to solve unknowns, like the rate of a second worker.

In the exercise, the equation \( \frac{1}{4} + r = \frac{2}{5} \) is set up where \( r \) represents the rate of the second proofreader. When solved, it reveals the missing information required to understand each person's contribution toward completing the entire task.

Reciprocal Rate Calculation

The reciprocal rate calculation is a technique used to uncover the time taken by a worker to finish a task alone. This involves taking the inverse of the work rate.

Once the second proofreader’s rate is calculated as \( \frac{3}{20} \) of the job per hour, the time to complete the task alone is found by taking its reciprocal. This means \( \frac{1}{\frac{3}{20}} \), simplifying to \( \frac{20}{3} \) hours, which equals 6 hours and 40 minutes. This reciprocal method allows us to convert the work rate into a tangible timeframe easily.