Two Printers Flyer Job: How Long For The Second Printer?

Hey guys! Ever wondered how to tackle a math problem involving two machines working together? Let's dive into a classic scenario: a company needs 10,000 flyers printed for a special offer. They've got two printers, and we need to figure out how long the second printer would take to do the job solo. Sounds like fun, right? This is a quintessential work-rate problem often encountered in mathematics, especially in algebra and arithmetic. These types of problems help us understand how different rates of work combine to achieve a common goal. So, buckle up, and let’s break this down step-by-step!

Understanding the Problem

Okay, so here’s the deal: we know one printer (Printer A) can handle all 10,000 flyers in 3 hours. We also know that if both printers (Printer A and Printer B) work together, they can finish the job in 1 hour and 40 minutes. The big question is: how long would it take Printer B to print all the flyers if it were working alone? To solve this, we need to think about the rate at which each printer works. Rate, in this context, means the fraction of the job completed per unit of time (in this case, per hour). Figuring out these individual rates is the key to unlocking the solution. We're essentially dealing with the concept of combined work rate, where the rates of multiple entities (in this case, printers) are added together to find their collective efficiency. This principle is widely applicable in various fields, from manufacturing to project management, where tasks are often distributed among different resources working concurrently.

Converting Time to a Common Unit

Before we start crunching numbers, let's make sure our time measurements are consistent. We have one time given in hours (3 hours) and another in hours and minutes (1 hour and 40 minutes). To make things easier, let’s convert everything to hours. We know that 40 minutes is 2/3 of an hour (since 40 minutes / 60 minutes per hour = 2/3). So, 1 hour and 40 minutes is the same as 1 + 2/3 hours, which equals 5/3 hours. This conversion is crucial because it allows us to perform calculations using a uniform unit of time. Failing to convert to a common unit can lead to inaccurate results and a misunderstanding of the problem. By expressing all time measurements in hours, we can directly compare and manipulate the rates of work for each printer.

Calculating the Work Rate of Printer A

Now, let's figure out how much of the job Printer A can do in one hour. If it takes Printer A 3 hours to complete the entire job (10,000 flyers), then in one hour, it completes 1/3 of the job. Think of it like this: the job is divided into three equal parts, and Printer A finishes one part each hour. This fraction, 1/3, represents the work rate of Printer A. Understanding work rate is fundamental to solving these types of problems. It allows us to quantify the amount of work done per unit of time, which is essential for comparing the efficiency of different workers or machines. In this case, knowing Printer A's work rate provides a baseline for understanding how the combined work rate of both printers contributes to the overall task completion.

Calculating the Combined Work Rate

Next up, let's figure out the combined work rate of both printers. We know they can complete the entire job in 5/3 hours. So, in one hour, working together, they complete 1 / (5/3) = 3/5 of the job. This means that if both printers are running, they finish 3/5 of the flyer printing in a single hour. The combined work rate is a critical concept because it represents the total output when multiple resources are working simultaneously. It's not simply the sum of the individual work rates; it takes into account the efficiency gained or lost when resources are combined. In this scenario, understanding the combined work rate allows us to determine how much faster the job gets done when both printers are operating compared to just one.

Finding the Work Rate of Printer B

This is where things get interesting! We know the combined work rate (3/5 of the job per hour) and the work rate of Printer A (1/3 of the job per hour). To find the work rate of Printer B, we simply subtract Printer A's work rate from the combined work rate. So, Printer B's work rate is (3/5) - (1/3). To subtract these fractions, we need a common denominator, which is 15. So, (3/5) becomes (9/15) and (1/3) becomes (5/15). Therefore, Printer B's work rate is (9/15) - (5/15) = 4/15 of the job per hour. Calculating the individual work rate of Printer B is crucial for answering the original question: how long would it take Printer B to complete the job alone? This step involves applying basic algebraic principles to isolate the unknown variable, which in this case is the rate of work for Printer B.

Calculating the Time for Printer B to Complete the Job Alone

Alright, we're in the home stretch! We know Printer B completes 4/15 of the job in one hour. To find out how long it takes to complete the entire job (1), we need to take the reciprocal of the work rate. So, the time it takes Printer B is 1 / (4/15) = 15/4 hours. Now, let's convert this improper fraction into a mixed number to make it easier to understand. 15/4 is equal to 3 and 3/4 hours. So, Printer B would take 3 and 3/4 hours to complete the job alone. Converting the work rate into the time taken to complete the job involves understanding the inverse relationship between rate and time. A higher work rate corresponds to a shorter time to completion, and vice versa. This principle is fundamental to solving work-rate problems and has practical applications in various real-world scenarios.

Converting Back to Hours and Minutes

Just to make it super clear, let’s convert 3 and 3/4 hours back into hours and minutes. We know we have 3 full hours, and 3/4 of an hour is 45 minutes (since (3/4) * 60 minutes = 45 minutes). So, Printer B would take 3 hours and 45 minutes to complete the job alone. This final conversion provides a more intuitive understanding of the time required for Printer B to complete the task. It also demonstrates the importance of being able to convert between different units of time, as this can often provide a clearer and more relatable answer to the problem.

Final Answer

So, there you have it! It would take the second printer, Printer B, 3 hours and 45 minutes to print all 10,000 flyers by itself. We did it! We successfully navigated a work-rate problem by breaking it down into smaller, manageable steps. Remember, the key to these problems is understanding the concept of work rate, converting time to a common unit, and carefully applying the principles of addition and subtraction. Whether you're dealing with printers, painters, or pipes filling a tank, the same fundamental approach will help you solve the puzzle. Practice makes perfect, so keep tackling these types of problems, and you'll become a work-rate whiz in no time! These problems not only enhance your mathematical skills but also develop your logical thinking and problem-solving abilities, which are valuable assets in various aspects of life.