How Long For The Second Pump To Fill The Tank A Work-Rate Problem

Hey guys! Ever found yourself scratching your head over a math problem that seems like it's speaking a different language? Well, today, we're diving headfirst into a classic work-rate problem that involves two pumps filling a tank. It's one of those questions that might seem daunting at first, but trust me, we're going to break it down into bite-sized pieces so that by the end, you'll be nodding along like a pro.

Cracking the Code: Understanding Work-Rate Problems

So, what exactly are work-rate problems? Work-rate problems are all about figuring out how long it takes individuals or, in our case, machines (like pumps) to complete a task, either working alone or together. The key here is to think about the rate at which work is being done. Think of it like this: if you can eat 2 slices of pizza per minute, your eating rate is 2 slices/minute. The same principle applies to pumps filling tanks, painters painting walls, or even writers writing articles. In essence, when tackling work-rate problems, it's essential to grasp that the rate at which work is completed is the pivotal element to consider. Understanding this foundational principle paves the way for effectively tackling and resolving such mathematical challenges.

When dealing with these problems, we often use a simple formula:

Work = Rate × Time

Where:

• Work is the amount of task completed (in our case, filling the tank, which we can consider as 1 whole tank).
• Rate is the speed at which the task is done (e.g., the fraction of the tank filled per hour).
• Time is the duration spent working on the task.

Imagine you are organizing a friendly race where participants are tasked with painting a fence. Each participant has a unique painting speed – some are naturally faster, while others may prefer a more leisurely pace. In this scenario, the "work" represents the entire fence that needs painting, the "rate" is the speed at which each participant can paint a section of the fence (measured in, say, meters per hour), and the "time" is the total duration each person spends painting. This analogy perfectly mirrors the work-rate problems we encounter in mathematics, where individuals or machines collaborate to accomplish a shared task. Grasping this concept makes it easier to develop strategies for solving these problems. Just as the combined efforts of the painters lead to the completion of the fence, understanding how individual work rates combine is key to unlocking solutions in various scenarios involving collaborative work.

Deconstructing the Problem: Two Pumps, One Tank

Okay, let's get down to the specific problem we're tackling today:

Two pumps working together can fill a tank in 12 hours. The first pump alone can fill the tank in 48 hours. How long would it take the second pump to fill the tank on its own?

This might sound like a mouthful, but don't worry, we're going to dissect it piece by piece. The first thing to do is identify the key information. We know the combined time for both pumps and the individual time for the first pump. What we don't know is the individual time for the second pump – that's our mystery to solve!

To truly break down this puzzle, let's delve deeper into the intricacies of how these pumps function and interact. Visualize the tank as a container gradually filling up, thanks to the concerted effort of both pumps. Each pump contributes its unique flow rate, working in tandem to achieve the common goal of filling the entire tank. Now, consider the first pump, which we know takes 48 hours to fill the tank on its own. This gives us a crucial piece of the puzzle: the individual work rate of the first pump. However, the second pump remains an enigma, its work rate yet to be deciphered. The question beckons – how long would it take this second pump, laboring in isolation, to accomplish the task? To answer this, we must embark on a journey of mathematical deduction. We'll need to leverage the information about their combined effort, along with the solitary contribution of the first pump, to unveil the hidden work rate of the second pump. This exploration promises to not only reveal the solution but also to deepen our understanding of the relationship between individual and collaborative work in these captivating work-rate problems.

Setting Up the Equation: Rate is the Key

Remember our Work = Rate × Time formula? This is where it comes in handy. Let's focus on the rates. If the first pump fills the tank in 48 hours, it fills 1/48 of the tank per hour. Similarly, if both pumps together fill the tank in 12 hours, they fill 1/12 of the tank per hour.

Let's use variables to make things even clearer:

• Let R1 be the rate of the first pump (1/48 tank per hour).
• Let R2 be the rate of the second pump (what we need to find).
• Let R_combined be the combined rate of both pumps (1/12 tank per hour).

Now, we can write an equation:

R1 + R2 = R_combined

This equation is the heart of our solution. It tells us that the sum of the individual rates of the pumps equals their combined rate. Think of it like this: each pump contributes a certain amount of effort per hour, and when you add those efforts together, you get the total effort exerted when they work in tandem. This concept of combining rates is fundamental in tackling work-rate problems. By understanding how individual contributions coalesce, we can effectively analyze and solve scenarios where multiple entities collaborate to achieve a common objective. It's a powerful principle that unlocks a wide array of mathematical puzzles, from filling tanks to completing complex tasks in the real world.

To truly grasp the significance of this equation, envision a team of chefs working together to prepare a grand feast. Each chef possesses a unique set of skills and works at their own pace, contributing specific dishes to the overall menu. In this analogy, the equation R1 + R2 = R_combined mirrors the combined efforts of the chefs. R1 represents the rate at which the first chef prepares their dishes, R2 symbolizes the rate of the second chef, and R_combined denotes the overall pace at which the feast is coming together. Just as the chefs' individual contributions blend seamlessly to create a culinary masterpiece, the equation harmonizes the rates of the pumps, allowing us to solve for the unknown work rate of the second pump. This analogy not only underscores the practical relevance of the equation but also vividly illustrates the collaborative nature of work-rate problems, where collective effort yields a common goal. With this understanding, we are well-equipped to navigate the complexities of the problem at hand and unveil the solution that lies within.

Solving for the Unknown: Unveiling the Second Pump's Speed

Now, let's plug in the values we know:

(1/48) + R2 = (1/12)

To solve for R2, we need to isolate it on one side of the equation. We can do this by subtracting (1/48) from both sides:

R2 = (1/12) - (1/48)

To subtract these fractions, we need a common denominator. The least common multiple of 12 and 48 is 48, so we can rewrite (1/12) as (4/48):

R2 = (4/48) - (1/48)

R2 = 3/48

Now, we can simplify the fraction by dividing both the numerator and denominator by 3:

R2 = 1/16

So, the second pump fills 1/16 of the tank per hour. This means it would take the second pump 16 hours to fill the entire tank on its own.

This journey of solving for the unknown work rate, R2, has been a testament to the power of mathematical deduction. Each step we've taken, from setting up the initial equation to simplifying the final fraction, has brought us closer to unraveling the mystery of the second pump's speed. Now that we've determined that the second pump fills 1/16 of the tank per hour, we can confidently assert that it would indeed require 16 hours for this pump to fill the entire tank on its own. This moment of realization is particularly satisfying because it demonstrates how the seemingly complex puzzle of work-rate problems can be systematically broken down and solved. The process of isolating the variable, finding common denominators, and simplifying fractions may seem like intricate steps, but they are the building blocks of mathematical problem-solving. By mastering these skills, we not only uncover the answer to a specific question but also gain a deeper appreciation for the elegance and precision of mathematics. The sense of accomplishment that comes from navigating these challenges reinforces the value of perseverance and analytical thinking, qualities that extend far beyond the realm of mathematics, influencing our approach to problems in all aspects of life.

The Big Reveal: 16 Hours to Tank-Filling Glory!

There you have it! The second pump, working solo, would take 16 hours to fill the tank. We successfully navigated this work-rate problem by focusing on rates, setting up a clear equation, and carefully solving for the unknown. Remember, guys, the key to tackling these problems is to break them down, understand the relationships between work, rate, and time, and don't be afraid to use variables to make things easier.

Pro Tips for Conquering Work-Rate Problems

Before we wrap up, let's arm you with some extra tips and tricks to become a true work-rate problem wizard:

1. Always think in terms of rates. This is the single most crucial aspect. Convert everything into