Snow Removal Time: Arthur, Andrey, And Dima's Yard Work

Let's dive into this snow removal problem involving Arthur, Andrey, and Dima! This is a classic type of math problem that involves understanding rates of work and how they combine when people work together. Guys, we're going to break down how to solve this step-by-step so you can tackle similar problems in the future. It might seem a bit tricky at first, but don't worry, we'll get through it together. The key here is to think about how much of the job each pair can do in a single unit of time, like one minute. Once we figure that out, we can combine their efforts and see how long it takes for all three of them to clear the snow together. So, grab your thinking caps, and let's get started on this snowy mathematical adventure! Remember, math isn't just about numbers; it's about problem-solving and logical thinking, skills that are super useful in all sorts of situations. So, let's get those mental shovels ready and clear away the confusion!

Understanding the Problem

The core of solving this snow removal problem lies in understanding the given information. We know the combined work rates of three pairs: Dima and Arthur, Arthur and Andrey, and Andrey and Dima. Specifically, Dima and Arthur can clear the yard in 45 minutes, meaning they complete 1/45 of the job per minute. Arthur and Andrey finish the job in 60 minutes, doing 1/60 of the work each minute. Lastly, Andrey and Dima can clear the snow in 90 minutes, completing 1/90 of the job per minute. The challenge is to find the combined work rate of all three individuals working together and, from that, determine the total time they take to clear the yard. This involves setting up a system of equations where each equation represents the combined work rate of a pair. By solving this system, we can find the individual work rates and then sum them to find the collective rate. This is a great example of how math can be used to model real-world situations, even something as simple as shoveling snow. So, let's dig into the equations and see how we can crack this problem!

Setting up the Equations

To solve this snow removal conundrum, we need to translate the given information into mathematical equations. Let's use variables to represent the fraction of the yard each person can clear in one minute: let 'A' be Arthur's rate, 'D' be Dima's rate, and 'N' be Andrey's rate. From the problem, we know three key pieces of information: Dima and Arthur together clear 1/45 of the yard per minute (A + D = 1/45), Arthur and Andrey clear 1/60 of the yard per minute (A + N = 1/60), and Andrey and Dima clear 1/90 of the yard per minute (N + D = 1/90). This gives us a system of three equations with three unknowns, which is solvable. These equations represent the combined work rates of each pair. The goal now is to solve this system to find the individual rates A, D, and N. This is a crucial step because once we know how much each person can do in a minute, we can easily figure out their combined effort when they all work together. So, let's put on our algebraic hats and start solving these equations!

Solving the System of Equations

Now comes the fun part: solving the system of equations we set up for the snow removal problem. We have three equations: (1) A + D = 1/45, (2) A + N = 1/60, and (3) N + D = 1/90. There are several ways to solve this system, but one common method is to use elimination or substitution. Let's use elimination. First, we can subtract equation (2) from equation (1) to eliminate 'A', giving us D - N = 1/45 - 1/60. Simplifying the right side, we get D - N = 1/180. Now we have a new equation along with equation (3): D - N = 1/180 and N + D = 1/90. We can add these two equations together to eliminate 'N', resulting in 2D = 1/180 + 1/90. Simplifying, we get 2D = 1/60, so D = 1/120. This means Dima can clear 1/120 of the yard in one minute. Next, we can substitute D's value back into equation (3) to find N: N + 1/120 = 1/90, so N = 1/90 - 1/120 = 1/360. Andrey clears 1/360 of the yard per minute. Finally, we substitute D's value back into equation (1) to find A: A + 1/120 = 1/45, so A = 1/45 - 1/120 = 1/72. Arthur clears 1/72 of the yard per minute. So, we've successfully found the individual work rates of Arthur, Dima, and Andrey!

Calculating the Combined Work Rate

With the individual work rates of Arthur, Andrey, and Dima determined, we can now calculate their combined snow removal rate. Remember, Arthur (A) clears 1/72 of the yard per minute, Dima (D) clears 1/120 per minute, and Andrey (N) clears 1/360 per minute. To find their combined rate, we simply add their individual rates together: Combined Rate = A + D + N = 1/72 + 1/120 + 1/360. To add these fractions, we need a common denominator, which in this case is 360. So, we rewrite the fractions: 1/72 = 5/360, 1/120 = 3/360, and 1/360 remains the same. Adding them together, we get: Combined Rate = 5/360 + 3/360 + 1/360 = 9/360. This simplifies to 1/40. Therefore, working together, Arthur, Andrey, and Dima can clear 1/40 of the yard in one minute. This is a key piece of information that will allow us to determine the total time it takes for them to clear the entire yard.

Determining the Total Time

Now that we know the combined work rate for snow removal, calculating the total time is the final step. We found that Arthur, Andrey, and Dima together clear 1/40 of the yard per minute. To find the total time it takes them to clear the entire yard, we need to find the reciprocal of their combined work rate. The reciprocal of 1/40 is 40/1, which is simply 40. This means it will take them 40 minutes to clear the entire yard when working together. So, guys, we've solved it! By understanding the individual and combined work rates, setting up equations, and doing some algebra, we've determined that these three friends can clear the snow in just 40 minutes if they team up. This problem highlights how math can be used to solve practical, real-world scenarios. Great job sticking with it, and remember, practice makes perfect when it comes to problem-solving!

Conclusion

In conclusion, we successfully tackled the snow removal problem, figuring out how long it would take Arthur, Andrey, and Dima to clear the yard together. By breaking down the problem into smaller parts, we were able to understand the individual work rates, set up a system of equations, solve for the unknowns, and ultimately find the combined work rate. This allowed us to determine that it would take them 40 minutes to clear the entire yard when working as a team. This exercise not only helps us appreciate the practical applications of mathematics but also reinforces the importance of problem-solving skills. Remember, many real-life situations can be modeled and solved using mathematical principles. So, keep practicing, keep thinking critically, and you'll be well-equipped to tackle any challenge that comes your way. Whether it's clearing snow, managing time, or any other task, a solid understanding of math can make all the difference. And remember, working together often makes the job easier and faster, just like Arthur, Andrey, and Dima showed us!