Wheel Rotation Calculation Step By Step Solution

by Scholario Team 49 views

Hey guys! Let's tackle this math problem together. It's all about figuring out how many rotations a wheel makes in a certain amount of time. We're given that a wheel completes 80 rotations in 20 minutes, and we need to find out how many rotations it will make in 28 minutes if it maintains the same speed. Sounds like a fun challenge, right? Let's dive in and break it down step by step.

Understanding the Problem

First, let's make sure we fully grasp what the problem is asking. We know the wheel's performance over a 20-minute period, and we need to predict its performance over a longer, 28-minute period. The key here is the phrase "maintaining the same speed." This tells us that the relationship between time and the number of rotations is consistent, which means we can use a proportion to solve this. Proportions are super handy for problems like these where things change at a constant rate.

What are we given?

  • The wheel makes 80 rotations.
  • This happens in 20 minutes.

What do we need to find out?

  • How many rotations will the wheel make in 28 minutes?

Now that we've clearly defined what we know and what we need to find, we can move on to setting up our solution. Remember, math problems are like puzzles – once you understand the pieces, putting them together is much easier. So, let's get those puzzle pieces in order!

Setting up the Proportion

The core of solving this problem lies in setting up a proportion. A proportion is simply a statement that two ratios are equal. In our case, the ratio we're dealing with is the number of rotations per minute. We know the wheel makes 80 rotations in 20 minutes, so we can express this as a ratio: 80 rotations / 20 minutes. This ratio represents the wheel's speed – how many rotations it makes for each minute that passes.

Now, we want to find out how many rotations the wheel will make in 28 minutes. Let's call the unknown number of rotations "x." We can set up another ratio: x rotations / 28 minutes. Since the wheel maintains the same speed, these two ratios must be equal. This gives us our proportion:

(80 rotations / 20 minutes) = (x rotations / 28 minutes)

This equation is the heart of our solution. It mathematically expresses the relationship between the known information (80 rotations in 20 minutes) and the unknown information (x rotations in 28 minutes). By solving for "x," we'll find the answer to our problem. Think of it like a balancing scale – the two sides of the equation must remain balanced. So, whatever we do to one side, we must do to the other.

With our proportion set up, we're now ready for the next step: solving for "x." This involves some simple algebraic manipulation, which we'll walk through carefully. Remember, the goal is to isolate "x" on one side of the equation, so we can see its value. Let's get to it!

Solving for 'x'

Alright, let's solve for 'x' in our proportion: (80 rotations / 20 minutes) = (x rotations / 28 minutes). To do this, we'll use a technique called cross-multiplication. Cross-multiplication is a handy way to solve proportions, and it's pretty straightforward.

Here's how it works:

  1. Multiply the numerator of the first fraction by the denominator of the second fraction: 80 rotations * 28 minutes.
  2. Multiply the denominator of the first fraction by the numerator of the second fraction: 20 minutes * x rotations.
  3. Set these two products equal to each other. This gives us a new equation without fractions, which is much easier to work with.

Let's apply this to our problem:

  • 80 * 28 = 2240
  • 20 * x = 20x
  • So, our new equation is: 2240 = 20x

Now we have a simple equation to solve for 'x'. To isolate 'x', we need to get rid of the 20 that's multiplying it. We can do this by dividing both sides of the equation by 20. Remember, whatever we do to one side, we must do to the other to keep the equation balanced.

  • 2240 / 20 = 112
  • 20x / 20 = x
  • So, we get: 112 = x

This means x = 112. We've found our answer! The wheel will make 112 rotations in 28 minutes. See? Solving for 'x' wasn't so scary after all. We just used a bit of cross-multiplication and some basic algebra.

Now that we've calculated the answer, let's make sure it makes sense in the context of the problem. This is an important step in problem-solving – it helps us catch any mistakes and ensures our answer is reasonable.

Verifying the Answer

Before we confidently say that 112 rotations is the final answer, let's take a moment to verify it. This means checking if our answer makes logical sense in the context of the problem. Think of it as a sanity check – we want to make sure we haven't made any silly mistakes along the way.

We know the wheel makes 80 rotations in 20 minutes. That's our starting point. We calculated that it would make 112 rotations in 28 minutes. Now, let's ask ourselves: does this make sense?

28 minutes is longer than 20 minutes, so we would expect the wheel to make more rotations. 112 rotations is indeed more than 80 rotations, so that's a good sign. But let's take it a step further. We can think about the increase in time and see if the increase in rotations is proportional.

The time increased by 8 minutes (28 minutes - 20 minutes = 8 minutes). The number of rotations increased by 32 rotations (112 rotations - 80 rotations = 32 rotations). Now, we can compare the ratio of the increase in rotations to the original ratio. If they're consistent, our answer is likely correct.

We can find the rotations per minute in both scenarios:

  • Original: 80 rotations / 20 minutes = 4 rotations per minute
  • New: 112 rotations / 28 minutes = 4 rotations per minute

The rotations per minute are the same in both cases! This strongly suggests that our answer of 112 rotations is correct. We've not only solved the problem mathematically, but we've also verified that the answer makes logical sense. This is a great habit to get into, as it builds confidence in your problem-solving abilities.

Final Answer and Conclusion

Okay, we've gone through all the steps – understanding the problem, setting up the proportion, solving for 'x', and verifying our answer. It's time to state our final conclusion. Based on our calculations, a wheel that makes 80 rotations in 20 minutes, while maintaining the same speed, will make 112 rotations in 28 minutes. That’s it! We've successfully solved the problem.

Looking back, we can see how breaking down the problem into smaller, manageable steps made it much easier to tackle. We identified the key information, set up a proportion to represent the relationship between time and rotations, used cross-multiplication to solve for the unknown, and verified our answer to ensure accuracy. These are valuable problem-solving skills that can be applied to many different situations.

So, the correct answer is A) 112 voltas. You nailed it! Remember, practice makes perfect, so keep working on these types of problems, and you'll become a math whiz in no time. And most importantly, don’t forget to have fun while you’re at it!

Is calculating wheel rotations causing you trouble? Discover a simple method to find the answer! We break down a typical question step-by-step in this article, assuring you comprehend the ideas and ace comparable mathematics problems. Math can be made easier, so let's get started!

The Original Problem

Let's restate the original problem to ensure everyone is on the same page. This will also assist readers who came searching for help with this specific problem via search engines such as Google. Here is the issue we're working on:

A wheel makes 80 rotations in 20 minutes. If the same speed is maintained, how many rotations will the wheel make in 28 minutes? Consider the alternatives: A) 112 rotations, B) 96 rotations, C) 104 rotations, D) 120 rotations. Calculate and justify your answer!

So, to reiterate, we need to figure out the relationship between time and wheel rotations. The main idea is that the wheel rotates at a consistent pace. This implies that the ratio of rotations to minutes stays constant. Understanding this basic concept is essential for successfully solving the problem. We'll go through each stage of the solution, ensuring you fully comprehend the reasoning.

Let's move on to the next step: properly deconstructing the issue. This entails identifying what we already know and what we need to find. Remember, good problem solving begins with a thorough understanding of the problem. We'll be one step closer to the solution once we've established the components.

Breaking Down the Problem

When faced with a math problem, the first step to success is breaking it down into smaller, more manageable parts. This makes the problem less intimidating and helps you identify the key pieces of information you need to solve it. Let's do that for our wheel rotation problem.

What do we know?

  • The wheel completes 80 rotations.
  • This happens in a time span of 20 minutes.
  • The wheel maintains a constant speed.

What do we need to find?

  • The number of rotations the wheel will make in 28 minutes.

By clearly stating what we know and what we need to find, we've created a roadmap for our solution. We know the relationship between rotations and time for one scenario (80 rotations in 20 minutes), and we want to use this relationship to predict the number of rotations in a different scenario (28 minutes). This is where the concept of proportions comes in handy.

Identifying Key Concepts

The key concept here is proportionality. Because the wheel maintains a constant speed, the ratio of rotations to time will be the same regardless of the time period. This allows us to set up a proportion, which is an equation stating that two ratios are equal. Proportions are a powerful tool for solving problems involving constant rates or relationships.

Now that we've broken down the problem and identified the key concepts, we're ready to move on to the next step: setting up the proportion. This is where we'll translate the word problem into a mathematical equation that we can solve. Stay tuned, and we'll get there together!

Setting Up the Proportion for Calculation

Now that we've understood the problem and identified the key information, it's time to translate that into a mathematical equation. This is where setting up a proportion comes in. A proportion, as we mentioned earlier, is an equation that states that two ratios are equal. In this case, we're dealing with the ratio of rotations to time.

We know that the wheel makes 80 rotations in 20 minutes. This gives us our first ratio: 80 rotations / 20 minutes. We want to find out how many rotations the wheel will make in 28 minutes. Let's represent the unknown number of rotations with the variable 'x'. This gives us our second ratio: x rotations / 28 minutes.

Since the wheel maintains a constant speed, these two ratios must be equal. This means we can set up the following proportion:

(80 rotations / 20 minutes) = (x rotations / 28 minutes)

This equation is the mathematical representation of our problem. It says that the ratio of rotations to minutes is the same in both scenarios. By solving for 'x', we'll find the number of rotations the wheel makes in 28 minutes. Isn't it amazing how we can express real-world situations using math?

Understanding Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics, and they're used in many different fields, from cooking to engineering. A ratio compares two quantities, while a proportion states that two ratios are equal. Understanding these concepts allows us to solve a wide range of problems involving relationships between quantities.

Now that we have our proportion set up, the next step is to solve for 'x'. This involves using some basic algebraic techniques, which we'll walk through step by step. Get ready to put your algebra skills to the test!

Calculating the Value of 'x' Step-by-Step

With our proportion set up as (80 rotations / 20 minutes) = (x rotations / 28 minutes), the next step is to solve for 'x'. This will give us the number of rotations the wheel makes in 28 minutes. We'll use a technique called cross-multiplication, which is a quick and easy way to solve proportions.

Cross-Multiplication Explained

Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. Then, we set these two products equal to each other. This eliminates the fractions and gives us a simpler equation to solve.

Let's apply cross-multiplication to our proportion:

  1. Multiply 80 rotations by 28 minutes: 80 * 28 = 2240
  2. Multiply 20 minutes by x rotations: 20 * x = 20x
  3. Set the two products equal to each other: 2240 = 20x

Now we have a linear equation: 2240 = 20x. To solve for 'x', we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 20:

  • 2240 / 20 = 112
  • 20x / 20 = x
  • So, we get: 112 = x

Therefore, x = 112. This means the wheel will make 112 rotations in 28 minutes. We've successfully calculated the value of 'x' using cross-multiplication and basic algebra. Give yourself a pat on the back!

Now that we have our answer, it's always a good idea to verify it. This helps us ensure that our answer makes sense in the context of the problem and that we haven't made any calculation errors. Let's move on to verifying our answer.

Confirming the Solution

Now that we've calculated that the wheel will make 112 rotations in 28 minutes, it's crucial to verify our solution. Verifying the solution helps us ensure that our answer is logical and accurate. There are a couple of ways we can do this.

Method 1: Checking Proportionality

We know that the wheel makes 80 rotations in 20 minutes. This means it makes 80 / 20 = 4 rotations per minute. If our answer is correct, the wheel should also make 4 rotations per minute when it makes 112 rotations in 28 minutes. Let's check:

  • 112 rotations / 28 minutes = 4 rotations per minute

The rotations per minute are the same in both scenarios, which confirms that our answer is proportional and likely correct.

Method 2: Logical Reasoning

We can also use logical reasoning to check our answer. 28 minutes is more than 20 minutes, so we would expect the wheel to make more rotations. 112 rotations is indeed more than 80 rotations, which supports our answer.

Additionally, we can think about the increase in time and the expected increase in rotations. The time increased by 8 minutes (28 - 20 = 8). Since the wheel makes 4 rotations per minute, we would expect an increase of 8 minutes * 4 rotations/minute = 32 rotations. Our answer shows an increase of 112 - 80 = 32 rotations, which aligns with our expectations.

Final Verification

Both methods confirm that our solution of 112 rotations is accurate and logical. This gives us confidence in our answer. Verifying your solutions is an essential step in problem-solving, as it helps you avoid errors and build a deeper understanding of the concepts.

The Final Answer

We've reached the final step of our problem-solving journey! We've carefully analyzed the problem, set up a proportion, solved for 'x', and verified our solution. Now it's time to state the final answer clearly and confidently.

Based on our calculations, the wheel will make 112 rotations in 28 minutes if it maintains the same speed. Therefore, the correct answer is A) 112 rotations.

We've successfully solved the problem! We took a seemingly complex question and broke it down into manageable steps. We used the concept of proportions, applied cross-multiplication, and verified our solution to ensure accuracy. This is a testament to the power of problem-solving skills.

Key Takeaways

  • Break down complex problems into smaller steps.
  • Identify key information and concepts.
  • Translate word problems into mathematical equations.
  • Use appropriate problem-solving techniques (like proportions and cross-multiplication).
  • Verify your solutions to ensure accuracy.

By following these steps, you can confidently tackle a wide range of math problems. Remember, practice makes perfect, so keep honing your skills and you'll become a math whiz in no time! And that wraps it up for this problem. Keep an eye out for more math adventures, and keep those wheels turning!

Do you want to calculate wheel rotations in a certain amount of time? This tutorial will show you how to solve this common math issue with ease. We go through the procedure step by step, making sure you grasp each notion. Math shouldn't be frightening; instead, let's make it simple!

Keywords Summary

To ensure that we've handled all of the critical search terms, let's go over the keywords associated with this problem:

  • Wheel rotations
  • Calculate rotations
  • Proportion problem
  • Solving for x
  • Verifying solutions
  • Step-by-step solution
  • Math problem
  • Time and rotations
  • Constant speed
  • 28 minutes
  • 80 rotations
  • 20 minutes

We've covered all of the major terms and subjects linked to the wheel rotation issue in this article. This assures that anyone looking for assistance with comparable math problems will find this information useful. Understanding these key phrases aids in grasping the problem's core ideas and efficiently using the solutions provided.

By methodically covering each area, we seek to give our readers a complete and simple understanding. Let's keep going and explore additional methods for making math fun and approachable!

Concluding Thoughts

So there you have it! We've successfully navigated the wheel rotation problem from start to finish. We've seen how breaking down a problem into smaller steps, identifying key concepts, and applying the right techniques can lead to a clear and accurate solution. Math doesn't have to be a mystery – with the right approach, it can be a rewarding and empowering skill.

Remember, the key to mastering math is practice. The more problems you solve, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're a natural part of the learning process. And always remember to verify your solutions to ensure accuracy.

Whether you're a student tackling homework or someone looking to brush up on your math skills, we hope this step-by-step solution has been helpful. Keep exploring, keep learning, and keep those wheels turning! Thanks for joining us on this math adventure.