Problem 42 - Elementary Algebra

Question

For Problems \(33-50\), set up an equation and solve the problem. (Objective 2 ) Betty can do a job in 10 minutes. Doug can do the same job in 15 minutes. If they work together, how long will it take them to complete the job?

Step-by-Step Solution

Answer

Together, it takes Betty and Doug 6 minutes to complete the job.

Step 1: Define Rates of Work

First, find the rate at which each person works. Since Betty can complete the job in 10 minutes, her rate is \( \frac{1}{10} \) of the job per minute. Doug's rate is \( \frac{1}{15} \) of the job per minute, as he completes the job in 15 minutes.

Step 2: Set Up the Combined Rate Equation

When working together, their combined rate of work will be the sum of their individual rates. Hence, the combined rate is \( \frac{1}{10} + \frac{1}{15} \). Let \( t \) be the time they take together to complete the job, so their combined rate can be expressed as \( \frac{1}{t} \).

Step 3: Solve for the Combined Rate

Add the rates of Betty and Doug to find their combined work rate: \[ \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \] Thus, the combined rate is \( \frac{1}{6} \), meaning together they complete \( \frac{1}{6} \) of the job per minute.

Step 4: Calculate the Time

Since their combined rate is \( \frac{1}{6} \), they take \( t \) minutes to complete the full job. Hence, \[ \frac{1}{t} = \frac{1}{6} \] Solving for \( t \), we find \( t = 6 \). This means that working together, they complete the job in 6 minutes.

Key Concepts

Rates of Work

When dealing with work problems in algebra, understanding the concept of "Rates of Work" is crucial. A rate of work indicates how much of a job is completed in a unit of time. For instance, if someone can finish a task in 10 minutes, their rate of work is expressed as \( \frac{1}{10} \) per minute. This means every minute, they complete one-tenth of the task.

It's helpful to think of rates of work similarly to speeds in motion problems. Just like a speed tells you how fast distance is covered, a rate of work tells you how quickly a task is completed. In our problem, Betty completes her job with a rate of \( \frac{1}{10} \) per minute, while Doug’s rate is \( \frac{1}{15} \) per minute.

Understanding these rates individually allows us to combine them when multiple workers collaborate, which brings us to the next key concept.

Combined Work Rate

When two or more people work together on the same task, their work rates combine to form a combined work rate. This is essential for calculating how fast they can complete the task as a team.

To determine the combined work rate, you simply add the individual rates together. In the example, Betty's rate is \( \frac{1}{10} \) of the job per minute, and Doug's rate is \( \frac{1}{15} \). Thus, their combined rate when working together is:

• \( \frac{1}{10} + \frac{1}{15} \)
• First, find a common denominator (which is 30 in this case)
• As a result, \( \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \)

This means that together, they complete \( \frac{1}{6} \) of the task every minute.

This collaborative rate gives us a basis to solve how long it will take them to finish the job together.

Solving Equations

After establishing the combined work rate, the next step is to solve the equation to find the total time required to complete the task together. Since the combined work rate is \( \frac{1}{6} \), this tells us how much of the job is completed per minute.

To find out the time \( t \) they need to complete the job, set their combined rate equal to \( \frac{1}{t} \):

• \( \frac{1}{t} = \frac{1}{6} \)

To solve for \( t \), take the reciprocal of both sides of the equation:

• \( t = 6 \)

This demonstrates that Betty and Doug together complete the job in 6 minutes. Solving equations in this format helps translate algebraic expressions into practical, real-world solutions.