Problem 39 - Elementary and Intermediate Algebra

Question

Printers. It takes a printer 6 hours to print the class schedules for all of the students enrolled in a community college. A faster printer can print the schedules in 4 hours. How long will it take the two printers working together to print \(\frac{3}{4}\) of the class schedules?

Step-by-Step Solution

Answer

It takes the two printers \(\frac{9}{5}\) hours, or 1.8 hours, to print \(\frac{3}{4}\) of the schedules.

Step 1: Define Work Rates

When working alone, the first printer can complete the entire job in 6 hours. Thus, its work rate is \(\frac{1}{6}\) of the job per hour. The second printer completes the job in 4 hours, so its work rate is \(\frac{1}{4}\) of the job per hour.

Step 2: Calculate Combined Work Rate

When both printers work together, their combined work rate is the sum of their individual work rates. Thus, the combined rate is \(\frac{1}{6} + \frac{1}{4}\). Convert these fractions to a common denominator: \(\frac{2}{12} + \frac{3}{12} = \frac{5}{12}\). Thus, together, they complete \(\frac{5}{12}\) of the job per hour.

Step 3: Determine Time to Complete 0 of Job

The problem asks for the time to complete \(\frac{3}{4}\) of the schedules. Using the combined work rate \(\frac{5}{12}\), set up the equation \(\frac{5}{12}x = \frac{3}{4}\). Solve for \(x\), where \(x\) is the time in hours.

Step 4: Solve the Equation for x

To solve \(\frac{5}{12}x = \frac{3}{4}\), first clear the fraction by multiplying both sides by 12: \(5x = 9\). Then divide each side by 5: \(x = \frac{9}{5}\). Thus, it takes \(\frac{9}{5}\) hours for the two printers to print \(\frac{3}{4}\) of the schedules.

Key Concepts

Rate of Work

Understanding the rate of work is crucial when solving problems that involve tasks being completed over time. Rates of work describe how much of a particular job an individual or machine can complete in a set unit of time, typically expressed as a part of the total job per hour. For instance, if a printer completes one job in 6 hours, its rate of work is \( \frac{1}{6} \) of the job per hour. Similarly, a second printer completing the same job in 4 hours has a rate of work of \( \frac{1}{4} \) of the job per hour.
Understanding these rates helps predict how much work can be completed both individually and collaboratively. When multiple entities work together, their combined rate of work is the sum of their individual rates. For example, the combined rate of two printers would be the sum of their individual work rates \( \frac{1}{6} + \frac{1}{4} \). This requires finding a common denominator to add the fractions effectively.

Fractions

Working with fractions is a key skill in many mathematical exercises, including work problems. Fractions represent parts of a whole, and in rate work problems, they indicate what portion of a job can be completed within a specific time frame. Understanding how to manipulate fractions is essential for combining work rates and solving for unknown variables.
Converting fractions to a common denominator is often necessary when adding or subtracting them. For example, to add \( \frac{1}{6} \) and \( \frac{1}{4} \), both fractions must be expressed with a common denominator, which in this case is 12, resulting in \( \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \). This sum represents the combined rate of work when two printers work together.
Solving work problems may also involve setting up and solving equations with fractions, such as solving \( \frac{5}{12}x = \frac{3}{4} \). This requires clearing the fraction by multiplying through by the denominator, and then isolating the variable on one side.

Algebra

Algebra plays a vital role in solving work problems as it provides the systematic procedures for manipulating equations and finding unknowns. In the context of this exercise, after determining the combined rate of work, you set up an equation to solve for the time it would take for both printers to complete a certain portion of the job.
With the combined work rate of \( \frac{5}{12} \), you would write the equation \( \frac{5}{12}x = \frac{3}{4} \) to find \( x \), the time required to complete \( \frac{3}{4} \) of the schedules. Solving this involves algebraic manipulation:

• Multiply through by 12 to clear the fraction, resulting in \( 5x = 9 \).
• Divide both sides by 5 to isolate x, giving \( x = \frac{9}{5} \), or alternatively 1.8 hours.

This step-by-step use of algebra allows for the clear resolution of the problem, showing not only the calculation but also the reasoning behind each step.