Problem 28 - Prealgebra and Introductory Algebra

Question

A manufacturer of prefabricated homes has the company's employees work in teams. Team 1 can erect the Silvercrest model in 15 h. Team 2 can erect the same model in 10 h. How long would it take for Team 1 and Team 2, working together, to erect the Silvercrest model home?

Step-by-Step Solution

Answer

Working together, Team 1 and Team 2 can erect the Silvercrest model home in 6 hours.

Step 1: Calculation of rates

The first step is to calculate the rates at which each team works. The rate of Team 1 is \( \frac{1}{15} \) houses per hour. For Team 2, the rate is \( \frac{1}{10} \) houses per hour.

Step 2: Adding the rates

The next step is to add these rates together. This will give the rate at which the teams can work collectively. This can be calculated using the formula: \( \frac{1}{15} + \frac{1}{10} = \frac{1}{6} \) houses per hour.

Step 3: Calculation of time

Finally, now that we have the combined rate of work, we can calculate how long it would take both teams to erect the Silvercrest model home. This is done by calculating the reciprocal of the combined rate, so that the time is given by: \( \frac{1}{\frac{1}{6}} = 6 \) hours.

Key Concepts

Combined Work Rates

Understanding combined work rates is essential when several workers or teams are collaborating on a single task, each contributing at different rates. In our exercise, we have two teams working together to build a house. The combined work rate is simply the sum of the individual rates of Team 1 and Team 2.

To effectively combine work rates, we must consider the work done by each team as a fraction of the total job completed per unit of time. For instance, if Team 1 erects a house in 15 hours, it means in one hour, they complete \(\frac{1}{15}\) of the house. Similarly, Team 2, which can complete the task in 10 hours, does \(\frac{1}{10}\) of the house per hour.

How to Add Work Rates

When we add these fractions together \( \frac{1}{15} + \frac{1}{10} \), we get the combined work rate. This represents the portion of the house both teams can build together in one hour. It is a simple yet powerful concept, enabling us to evaluate the efficiency of collaborative work. By summing individual rates, we can resolve how much time the combined effort will save.

Reciprocal of Rates

The reciprocal of a work rate is a fundamental concept when dealing with rate problems. The reciprocal inverts the relation between the quantity of work done and the time taken to do it.

For example, if a team has a work rate of \(\frac{1}{10}\) of the home per hour, the reciprocal would be the amount of time it takes to complete one whole home, which in this case would be 10 hours. It's like turning the problem on its head: instead of looking at what fraction of a task can be done in an hour, we look at how many hours it would take to complete the entire task.

Applying the Reciprocal to Combined Work Rates

After obtaining the combined work rate of the teams, we must find the reciprocal to ascertain the total time to complete one whole house collaboratively. This transformation is vital for translating a collective work rate back into a tangible time estimate for the entire task.

Work Rate Algebra

Work rate algebra is the mathematical manipulation of work rates to solve for unknown variables, commonly time or efficiency of workers. It applies algebraic techniques to equations derived from work scenarios, and is highly useful in combining rates and solving complex work problems.

The crux of work rate algebra lies in the idea that the rate of work times time equals work done. To illustrate, if Team 1 has a work rate of \(\frac{1}{15}\) houses per hour, then multiplying this rate by the number of hours worked gives us the total houses built by Team 1. When we consider multiple teams or workers, their rates can be added algebraically to find a collective rate.

Formulating Solutions

After establishing the collective work rate, algebra can then be used to rearrange the formula and solve for the desired variable, such as total time. In our exercise, we identified the combined work rate and used the reciprocal operation, a critical aspect of work rate algebra, to ascertain the time required for both teams to complete the Silvercrest model home.