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Chapter 1: Problem 80

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?

Short Answer

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

Step by step solution

01

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 \( \frac{1}{25} \)
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve the inequality. Then graph the solution set on the real number line. \(-10 x<40\)

You stop at a self-service gas station to buy 15 gallons of 87 -octane gasoline at \(\$ 2.42\) a gallon. The pump scale is accurate to within one-tenth of a gallon. How much money might you have been undercharged or overcharged?

The average professional baseball player's salary \(S\) (in millions of dollars) from 1995 to 2006 can be modeled by \(S=0.1527 t+0.294, \quad 5 \leq t \leq 16\) where \(t\) represents the year, with \(t=5\) corresponding to 1995 (see figure). Use the model to predict the year in which the average professional baseball player's salary exceeds \(\$ 3,000,000\). (Source: Major League Baseball)

The average price \(G\) (in dollars) of generic prescription drugs from 1998 to 2005 can be modeled by \(G=2.005 t+0.40, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to \(1998 .\) Use the model to find the year in which the price of the average generic drug prescription exceeded \(\$ 19\).

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). An equation that models the data you obtained is \(y=1.4 x-39\) (a) Estimate the values of \(x\) that predict a maximum bench-press weight of at least 200 pounds. (b) Do you think an athlete's weight is a good indicator of the athlete's maximum bench-press weight? What other factors might influence an individual's bench-press weight?
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