Problem 12 - Intermediate Algebra

Question

Decorating. One crew can put up holiday decorations in a department store in 12 hours. A second crew can put up the decorations in 15 hours. How long will it take if both crews work together to decorate the store?

Step-by-Step Solution

Answer

Both crews together take approximately 6.67 hours to complete the job.

Step 1: Determine the Work Rate of Each Crew

The first crew can complete the entire decoration job in 12 hours. Therefore, their work rate is \( \frac{1}{12} \) of the job per hour. The second crew can complete the job in 15 hours, so their work rate is \( \frac{1}{15} \) of the job per hour.

Step 2: Calculate the Combined Work Rate

To find the combined work rate of both crews working together, add their individual work rates: \( \frac{1}{12} + \frac{1}{15} \). Find a common denominator (which is 60): \( \frac{5}{60} + \frac{4}{60} = \frac{9}{60} \). Simplify this fraction to \( \frac{3}{20} \). This is the rate at which both crews work together.

Step 3: Solve for the Time Working Together

If both crews together complete \( \frac{3}{20} \) of the job in one hour, the time \( t \) it takes to complete the whole job is given by solving \( \frac{3}{20}t = 1 \). Therefore, \( t = \frac{20}{3} \), which is approximately 6.67 hours.

Key Concepts

Understanding the Combined Work Rate

When tackling problems involving work rates, especially when multiple parties work together, it's crucial to determining their combined efficiency. The combined work rate is the sum of the work rates of all parties involved.
For instance, if Crew A can complete a job in 12 hours, their work rate is \( \frac{1}{12} \) of the job per hour. Similarly, if Crew B takes 15 hours, their work rate is \( \frac{1}{15} \) of the job per hour.

• This means Crew A completes one-twelfth and Crew B completes one-fifteenth of the job per hour.
• The combined work rate is simply the sum of these individual rates, so you add \( \frac{1}{12} \) and \( \frac{1}{15} \).

By finding a common denominator and adding these fractions, you calculate the rate at which they work together.

Algebraic Problem Solving Techniques

Algebra is a powerful tool to solve work rate problems efficiently. You need to employ some basic algebraic techniques to calculate times or rates. The essence lies in setting up equations that represent the situation you are analyzing.

• First, identify the individual work rates. Assign these as fractions, representing parts of the job done in a specific time.
• Next, set up an equation to represent their combined effort. This equation typically involves summing the fractions and equating them to represent a whole job done.

For example, once you find the combined rate \( \frac{3}{20} \), setting up the equation \( \frac{3}{20}t = 1 \) helps find how long it takes to complete the job together. Solve for \( t \) to find the total time.

Calculating Task Completion Time

Once you determine the combined work rate, the next step is to find the task completion time. This represents the total time both parties need when working together.
This is done by dividing the complete job, represented as '1', by the combined work rate.

• Use the equation \( \text{work rate} \times \text{time} = 1 \) to solve for time.
• Rearrange this to \( \text{time} = \frac{1}{\text{work rate}} \), where the work rate is the combined rate found earlier.

In our example, dividing 1 by \( \frac{3}{20} \) gives approximately 6.67 hours, demonstrating how efficient the crews are when cooperating. This method ensures a step-by-step calculation for better understanding.