Problem 31 - Elementary and Intermediate Algebra

Question

Roofing Houses. A homeowner estimates that it will take her 7 days to roof her house. A professional roofer estimates that he could roof the house in 4 days. How long will it take if the homeowner helps the roofer?

Step-by-Step Solution

Answer

It will take them approximately 2.55 days to roof the house together.

Step 1: Determine Rates of Work

First, we need to find out how much of the house each can roof in one day. The homeowner takes 7 days alone, so she can complete \( \frac{1}{7} \) of the roof in one day. The professional roofer takes 4 days alone, so he can complete \( \frac{1}{4} \) of the roof in one day.

Step 2: Combine Rates of Work

To determine how much they can complete together in one day, add their rates of work: \( \frac{1}{7} + \frac{1}{4} = \frac{4}{28} + \frac{7}{28} = \frac{11}{28} \). Thus, together they can complete \( \frac{11}{28} \) of the roof in one day.

Step 3: Calculate Total Time Needed

Now that we know together they complete \( \frac{11}{28} \) of the roof in one day, set up the equation for one whole roof: \[ \frac{11}{28}t = 1 \]. Solve for \( t \) by multiplying both sides by \( \frac{28}{11} \): \[ t = \frac{28}{11} \].

Step 4: Convert Fraction to Days

Simplify \( \frac{28}{11} \): it equals approximately 2.55 days, or 2 days and about 13 hours. Therefore, working together, they will complete the roofing in approximately 2.55 days.

Key Concepts

Collaborative Work

Collaborative work involves two or more individuals coming together to achieve a common goal. In the context of work problems, it often means combining efforts to complete a task faster than each could individually. For example, when both the homeowner and the professional roofer work on the roof together, they share their skills and time to complete the job more efficiently. This collaboration can dramatically speed up the project because both parties are working simultaneously. Benefits of collaborative work include:

• Improved efficiency by dividing tasks based on expertise.
• Reduced time for project completion compared to working alone.
• Enhanced learning as individuals share their knowledge and skills.

In our exercise, by collaborating, they finish roofing in a little over 2.5 days, much faster than either could alone.

Work Rate Problems

Work rate problems are commonly part of algebra exercises that focus on determining how long it takes various workers to complete a task, either alone or together. The key to solving these problems is understanding the concept of 'rate of work', which is how much of a task can be completed in a specific time period. In the roofing problem, each individual's work rate is calculated as the fraction of the task completed in one day. For instance, if a roofer can complete the entire roof in 4 days, their work rate is \( \frac{1}{4} \) of the roof per day. Comparing and combining different rates of individuals leads us to compute how they can expedite work when their efforts are combined. Understanding these problems helps in:

• Evaluating and managing time efficiently in project planning.
• Balancing workloads among team members.
• Optimizing resources for better productivity.

Algebraic Equations

Algebraic equations form the backbone of calculating work problems, where we use them to find unknowns, such as the time required to complete a task. In our scenario, the equation \( \frac{11}{28}t = 1 \) helps determine how long the homeowner and roofer take in combination to finish roofing the house. Here, the term \( \frac{11}{28} \) is the combined rate of work per day, and the variable \( t \) represents time in days. Solving this equation involves isolating \( t \) to find the total number of days, which is done by multiplying both sides by the reciprocal of the rate, \( \frac{28}{11} \), leading to \( t = \frac{28}{11} \), or roughly 2.55 days. The use of algebra in these problems allows:

• Precise calculation of time based on known rates.
• Adjustment of variables to understand potential changes in work conditions.
• Logical structure to approach and solve real-world time and labor issues.