Problem 5 - Elementary and Intermediate Algebra

Question

a. It takes a night security officer 45 minutes to check each of the doors in an office building to make sure they are locked. What is the officer's rate of work? b. It takes an elementary school teacher 4 hours to make out the semester report cards. What part of the job does she complete in \(x\) hours?

Step-by-Step Solution

Answer

Officer's rate: \(\frac{4}{3}\) doors/hour; teacher completes \(\frac{x}{4}\) of the job in \(x\) hours.

Step 1: Define the Total Time and Task

For each situation, we need to identify the total time it takes to complete the task and the task itself. - For the security officer, it takes 45 minutes to check all the doors. - For the teacher, it takes 4 hours to complete the report cards.

Step 2: Determine the Rate of Work for the Officer

The officer's rate of work can be calculated as the reciprocal of the total time it takes him to complete the task. Hence, the rate of work is given by \[\text{Rate of work for officer} = \frac{1}{\text{total time in hours}}\]Converting 45 minutes to hours: \[45 \text{ minutes} = \frac{45}{60} = 0.75 \text{ hours}\]Hence, the officer's rate of work is \[\frac{1}{0.75} \text{ doors per hour} = \frac{4}{3}\text{ doors per hour}\]

Step 3: Determine the Part of the Job the Teacher Completes in x Hours

The teacher completes a job in a known total time of 4 hours, which means her rate is \[\text{Rate of work for teacher} = \frac{1}{4} \text{ of the job per hour}\]Therefore, if the teacher works for \( x \) hours, the part of the job completed is \[\text{Part completed} = x \times \frac{1}{4} = \frac{x}{4}\]

Step 4: Conclusion

We have calculated that:- The officer works at a rate of \( \frac{4}{3} \) doors per hour.- The teacher completes \( \frac{x}{4} \) of the report cards in \( x \) hours.

Key Concepts

Conversion of Units

Understanding how to convert units is essential in solving work rate problems. Often, the initial information given is not in the most convenient units for calculation. For example, in our problem, the night security officer's time is given in minutes, whereas the rate of work is more meaningfully expressed in hours. To convert 45 minutes into hours, we need to recognize the relationship between minutes and hours:

• 1 hour = 60 minutes.
• To express 45 minutes in hours, divide 45 by 60.

Thus, \[\frac{45}{60} = 0.75 \text{ hours}.\]This conversion allows us to calculate rates consistently in terms of hours, providing a standard measure for analysis. Being comfortable with such conversions is crucial to mastering any work rate calculation.

Fraction Multiplication

Fraction multiplication is a key skill when dealing with rates and parts of a job completed. In determining how much work the elementary school teacher accomplishes in a given time, we multiply her rate per hour by the number of hours she works. Her rate of \(\frac{1}{4}\) represents completing one-fourth of work per hour.

• To find the portion of the job completed in \(x\) hours, we multiply the rate by \(x\):
• \(x \times \frac{1}{4} = \frac{x}{4},\) which shows that the teacher completes \(\frac{x}{4}\) parts of the total work.

Each term is multiplied straightforwardly, emphasizing the ease and efficiency of computations involving fractions.

Reciprocal Calculation

Reciprocal calculation is applied to work rate problems to simplify the expression of rates. When we consider the work rate of the night security officer, we express it as the reciprocal of the total time taken to complete one job. By calculating this, we find that the rate is given by:

• \(\text{Rate of work for officer} = \frac{1}{0.75} = \frac{4}{3} \) doors per hour.

Reciprocals invert the fraction, making them crucial in translating between total time and rate of work. Understanding reciprocals helps reveal the full picture of repetitive tasks, providing a clear measure to compare the efficacy of different workers or processes.