Problem 104 - A Graphical Approach to College Algebra

Question

Solve each problem involving rate of work. Tommy and Alicia are laying a tile floor. Working alone, Tommy can do the job in 20 hours. If the two of them work together, they can complete the job in 12 hours. How long would it take Alicia to lay the floor working alone?

Step-by-Step Solution

Answer

Alicia would take 30 hours to complete the job alone.

Step 1: Define Variables for Rate of Work

Let's define Tommy's work rate and Alicia's work rate. Since Tommy can complete the job in 20 hours on his own, his rate of work is \( \frac{1}{20} \) of the job per hour. Let \( x \) be the number of hours it takes Alicia to complete the job alone. Then her rate of work is \( \frac{1}{x} \) of the job per hour.

Step 2: Set Up the Equation for Combined Work

When Tommy and Alicia work together, they can complete the job in 12 hours. This means their combined work rate is \( \frac{1}{12} \) of the job per hour. So, the equation for their combined work is: \( \frac{1}{20} + \frac{1}{x} = \frac{1}{12} \).

Step 3: Solve the Equation for Alicia's Work Rate

To find \( x \), solve the equation \( \frac{1}{20} + \frac{1}{x} = \frac{1}{12} \). Subtract \( \frac{1}{20} \) from both sides to isolate \( \frac{1}{x} \):\[ \frac{1}{x} = \frac{1}{12} - \frac{1}{20} \].

Step 4: Simplify and Solve for \( x \)

Find a common denominator for \( \frac{1}{12} \) and \( \frac{1}{20} \), which is 60. Rewriting both fractions: \( \frac{1}{12} = \frac{5}{60} \), and \( \frac{1}{20} = \frac{3}{60} \). Now subtract them: \( \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30} \).So, \( \frac{1}{x} = \frac{1}{30} \), which means \( x = 30 \).

Step 5: Conclusion on Alicia's Time

Thus, Alicia working alone would take 30 hours to complete the job.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is about finding the unknown or putting real-life variables into equations and then solving them.

In rate of work problems like the one we have here, algebra is essential because it allows us to set up expressions that represent real-world situations. For example, we represent the rate at which Tommy can perform his task as \( \frac{1}{20} \). This is algebra at work, allowing us to simplify complex problems into manageable equations.
When solving such problems, our goal is to find key information by isolating variables. Algebraic manipulation helps achieve that. In this specific problem, we solve for the time Alicia would take to complete a job using these principles of algebra.

Work Rate Equation

Work rate equations help us understand how different entities contribute to completing a collective task. This is crucial in determining the time it takes for individuals working together to achieve something.

The basic formula for work rate problems is:

• Work Done = Rate of Work × Time

In the problem at hand, we know:

• Tommy's work rate is \( \frac{1}{20} \) (because he can complete the job alone in 20 hours).
• Alicia's work rate is unknown and represented as \( \frac{1}{x} \).

When combined, the total work rate is \( \frac{1}{12} \) since they can finish together in 12 hours. By setting up the equation \( \frac{1}{20} + \frac{1}{x} = \frac{1}{12} \), we can solve for Alicia’s work rate. Understanding this equation is key to tackling the problem effectively.

Problem Solving Steps

Tackling rate of work problems systematically involves several clear steps to ensure precision in the calculations.

The process begins with defining the variables for work rates. You start by understanding individual contributions, as in defining \( \frac{1}{20} \) for Tommy. Then, you create an expression such as \( \frac{1}{x} \) for Alicia reflecting her unknown contribution.
Setting up the equation is next, combining their hourly work rates. The equation \( \frac{1}{20} + \frac{1}{x} = \frac{1}{12} \) comes from their shared work, clarifying how they together finish the job. The goal is to simplify and rearrange the equation to isolate the unknown variable.
Finally, simplifying requires calculating a common denominator and performing arithmetic to solve for \( x \). This methodological approach turns complex word problems into manageable mathematical solutions.

Combining Work Rates

Combining work rates involves adding the individual rates of people working towards a singular goal. This concept assumes that work done by different parties can be summed linearly.

For Tommy and Alicia, we express their combined effort as separate work contributions added together. Tommy’s rate is \( \frac{1}{20} \), and Alicia’s unknown contribution is \( \frac{1}{x} \). Together, their combined rate is \( \frac{1}{12} \), the rate at which they complete the task.
By equating \( \frac{1}{20} + \frac{1}{x} = \frac{1}{12} \), we use a strategy that places both individuals' input into a single framework. Solving this framework clarifies the balance of their activities, capturing how their joint efforts complete the problem's task efficiently. Understanding this helps students appreciate the collaborative nature of shared tasks and efficient solutions.