Problem 80 - College Algebra and Calculus: An Applied Approach

Question

You and a friend volunteer to paint a large house as a community service project. Working alone, you can paint the house in 28 hours. Your friend can paint the house in 25 hours working alone. How long will it take both of you, working together, to paint the house?

Step-by-Step Solution

Answer

It would take both of you approximately 13 hours (rounded to the nearest whole number) to paint the house when working together.

Step 1: Finding Individual Work Rates

Let's find out the work rates. The work rate is given by the formula \( \text{work rate} = \frac{1}{\text{time}} \). Here, your work rate would be \( \frac{1}{28} \) and your friend's work rate would be \( \frac{1}{25} \)

Step 2: Calculating Combined Work Rate

The combined work rate when two or more people are working together is the sum of their individual work rates. So, the work rate when you and your friend work together is \( \frac{1}{28} + \frac{1}{25} \). Simplify this to get the combined work rate.

Step 3: Calculating Time to Complete the Job Together

Finally, to find out how long it will take to complete the job together, use the formula \( \text{time} = \frac{1} {\text{work rate}} \). Substitute the combined work rate from Step 2 into this formula to get the time it would take both of you to paint the house together.

Key Concepts

Individual Work Rate

Understanding individual work rate is foundational in solving work and time problems. Imagine each person working alone on a task, like painting a house. The individual work rate tells us the portion of the job completed per unit of time. This is calculated using the formula \( \text{work rate} = \frac{1}{\text{time}} \).

This means if someone can complete a task in 28 hours alone, their rate is \( \frac{1}{28} \) of the job per hour. It's like saying, "In one hour, I complete \( \frac{1}{28} \) of the house." If your friend can do the same task in 25 hours, their rate is \( \frac{1}{25} \).
Knowing this makes it easy to calculate how much work is done by each individual over any period.

Combined Work Rate

When two people work together, we calculate their combined work rate by adding their individual work rates. This is crucial for understanding how teamwork speeds up the completion of tasks. For instance, if you and your friend want to paint a house together, your combined work rate is \( \frac{1}{28} + \frac{1}{25} \).

This represents how much of the job you both complete in one hour together. Simplifying the sum of these fractions gives us the total work done per hour.

• Firstly, find a common denominator to add the fractions.
• Secondly, add the numerators.
• Finally, simplify the result.

The combined work rate is essential in figuring out how much time it will take for both of you to finish the job when collaborating.

Problem-Solving in Algebra

Problem-solving with algebra comes in handy when calculating work and time, especially combining individual efforts. When two people work together, use algebra to find the time to complete the job. Start with the combined work rate derived previously. The formula for time is given by \( \text{time} = \frac{1}{\text{work rate}} \).

For the painting scenario, substitute your calculated combined work rate into this formula. It will give you the result for how long it will take to complete the house together.

• Recognize the importance of simplification and substitution.
• Ensure you convert fractions correctly for accurate results.
• This method highlights the power of algebra in everyday tasks.

Understanding fractional work rates and the algebraic formula helps solve complex real-world problems efficiently.