Problem 14 - Elementary Algebra Within Reach

Question

Typing Project One person can complete a typing project in 6 hours, and another can complete the same project in 8 hours. How long will it take the two people working together to complete the project?

Step-by-Step Solution

Answer

It will take approximately 3.43 hours for the two people working together to complete the project.

Step 1: Identify the individual work rates

First, identify the rate at which each person can complete the work individually. The rate is defined as the amount of work that can be done per unit time. Here, the first person completes the project in 6 hours, thus their rate is \(1/6\) project per hour. Similarly, the second person completes the project in 8 hours, so their rate is \(1/8\) project per hour.

Step 2: Calculate combined work rate

The total rate of the two people working together is the sum of their individual rates. Hence, the combined work rate is \((1/6 + 1/8)\) projects per hour.

Step 3: Find the time taken to complete the project

To find the time taken to complete the project, divide the total work (one project) by their combined work rate. That is \(1 / (1/6 + 1/8)\) hours.

Key Concepts

Individual Work Rate

When we speak about 'individual work rate,' we're looking at how much work a single person can do in a certain time period. For the given problem, it's all about how quickly each of our two people can complete a typing project on their own.

This is expressed as a fraction, representing the part of the work done in one hour.

• For the first person, completing the project in 6 hours means they do \( \frac{1}{6} \) of the project per hour.
• For the second person, completing it in 8 hours gives them a rate of \( \frac{1}{8} \) of the project per hour.

Understanding individual work rate lays the foundation for solving work rate problems, as it allows us to see each person's contribution to the task. This concept is further used to calculate how these contributions add up when working as a team.

Combined Work Rate

Once we know each person's individual work rate, the next step is to find out how fast they can complete the project together, which is their 'combined work rate.'

In this context, the combined work rate is simply the sum of the individual work rates.
For our typists:

• Their combined rate is \( \frac{1}{6} + \frac{1}{8} \) of the project per hour.
• This requires us to calculate \( \frac{1}{6} + \frac{1}{8} \), which involves finding a common denominator to add these fractions.
• Once we have a common denominator, we can add these fractions to find their combined rate in projects per hour.

This concept is essential because it determines how efficiently a team can work together, based on the speeds of individual workers. By adding up the rates, we get a clear picture of their collaboration potential.

Time Taken to Complete Work

The final piece of the puzzle in work rate problems is determining the total time required to finish the job when working together—this is the 'time taken to complete work.'

We find this by diving into a simple division operation: taking the whole project (considered as 1 unit of work) and dividing it by the combined work rate.

• The formula here is \( \text{Time} = \frac{1}{\text{Combined Work Rate}} \).
• For example, with the combined rate from our earlier step, \( \frac{1}{6} + \frac{1}{8} \), the time it takes is \( \frac{1}{(\frac{1}{6} + \frac{1}{8})} \).
• After calculating the combined rate, we substitute this into the formula to find the total time they need.

This step concludes the process, providing the concrete duration for the teamwork to succeed. It transforms individual efforts into a composite outcome, illustrating how work can be completed efficiently when resources are pooled together.