Solving R = (13/5) × (8/7) × (75/39) × (91/64) A Step By Step Guide

Hey everyone! Today, we're diving into a fun math problem. We're going to break down the equation R = (13/5) × (8/7) × (75/39) × (91/64) step by step so you can see exactly how to solve it. Math can seem intimidating at first, but trust me, when you take it piece by piece, it becomes much more manageable—and even kind of fun! So, grab your pencils and let’s get started on this mathematical adventure together! We will explore each fraction, simplify where possible, and then bring it all together to find the value of R. Are you ready? Let's jump right into it!

Understanding the Equation

Before we start crunching numbers, let’s make sure we understand what the equation is telling us. The equation we're tackling is R = (13/5) × (8/7) × (75/39) × (91/64). What we're looking for here is the value of R, which is the result of multiplying these four fractions together. Think of it like this: we have four different pieces, and we need to combine them in the right way to get our final answer. Each fraction represents a part of the whole multiplication problem, and when we multiply them, we're essentially finding a fraction of a fraction of a fraction, and so on. This might sound a bit complex, but don't worry! We'll break it down so it’s super clear.

Now, why is understanding this important? Well, it’s like having a map before you start a journey. If you know where you’re going, the trip is much smoother. In math, understanding the problem sets you up for success. We need to see the structure of the equation so we can plan our attack. We have fractions being multiplied, so we know we can simplify before multiplying to make our lives easier. Simplifying fractions means reducing them to their lowest terms, which keeps the numbers smaller and more manageable. This step is crucial because it prevents us from dealing with huge numbers that can be hard to handle. So, with our map in hand, let’s start simplifying these fractions and making our journey to the solution much easier. Trust me; you'll be surprised how much simpler it becomes once we've trimmed down those numbers!

Step 1: Simplify the Fractions

Okay, guys, the first real step in solving our equation R = (13/5) × (8/7) × (75/39) × (91/64) is to simplify each fraction as much as possible. Simplifying fractions makes the multiplication much easier because we're dealing with smaller numbers. It's like decluttering your workspace before starting a big project; it just makes everything cleaner and more efficient. So, let’s go through each fraction one by one and see what we can simplify. We're looking for common factors between the numerator (the top number) and the denominator (the bottom number) in each fraction. Remember, a common factor is a number that divides evenly into both the numerator and the denominator.

First up, we have 13/5. Can we simplify this? Well, 13 is a prime number, which means it’s only divisible by 1 and itself. And 5 is also a prime number, only divisible by 1 and itself. Since they don't share any common factors other than 1, 13/5 is already in its simplest form. Nothing to do here but move on to the next one! Next, we have 8/7. Again, 7 is a prime number, and the factors of 8 are 1, 2, 4, and 8. They don’t share any common factors, so 8/7 is also in its simplest form. We’re on a roll! Now let’s look at 75/39. This one looks promising for simplification. The factors of 75 are 1, 3, 5, 15, 25, and 75, while the factors of 39 are 1, 3, 13, and 39. Aha! We see a common factor: 3. So, we can divide both 75 and 39 by 3. 75 divided by 3 is 25, and 39 divided by 3 is 13. That means 75/39 simplifies to 25/13. Awesome! One down, one to go. Finally, we have 91/64. The factors of 91 are 1, 7, 13, and 91, and the factors of 64 are 1, 2, 4, 8, 16, 32, and 64. We spot a common factor here as well: there isn't any! So 91/64 will remain as it is. By simplifying, we’ve made the equation much easier to handle. We’ve transformed 75/39 into the simpler 25/13, which will make the next steps less cumbersome. Simplifying fractions is a crucial skill in math, and it’s going to save us a lot of time and effort in the long run. Now that we’ve simplified as much as possible, we can move on to the next step: multiplying the simplified fractions.

Step 2: Multiply the Simplified Fractions

Alright, now that we've simplified our fractions, it’s time to multiply them together. Our equation now looks like this: R = (13/5) × (8/7) × (25/13) × (91/64). Remember, when we multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we're going to multiply 13, 8, 25, and 91 to get the new numerator, and we'll multiply 5, 7, 13, and 64 to get the new denominator. This might seem like a big task, but don't worry, we'll take it step by step, and we'll look for opportunities to simplify along the way. Multiplying all these numbers out directly can lead to some pretty large numbers, which can be a bit unwieldy. Instead, let's use a clever trick: we can simplify before we multiply. This means we look for common factors between any numerator and any denominator and cancel them out. It’s like pre-simplifying our big multiplication problem. So, let’s write out our multiplication like this: (13 × 8 × 25 × 91) / (5 × 7 × 13 × 64).

Now, let’s start canceling out common factors. Do you see any numbers on the top and bottom that can divide into each other? I spot a 13 on both the top and the bottom. We can cancel those out because 13 divided by 13 is 1. So, we’re left with (1 × 8 × 25 × 91) / (5 × 7 × 1 × 64). What else can we simplify? I see an 8 in the numerator and a 64 in the denominator. Both are divisible by 8. 8 divided by 8 is 1, and 64 divided by 8 is 8. Now we have (1 × 1 × 25 × 91) / (5 × 7 × 1 × 8). We’re making good progress! Next, let’s look at the 25 in the numerator and the 5 in the denominator. Both are divisible by 5. 25 divided by 5 is 5, and 5 divided by 5 is 1. Our equation is now (1 × 1 × 5 × 91) / (1 × 7 × 1 × 8). See how much simpler this is getting? We have one more nice simplification we can make. Look at the 91 in the numerator and the 7 in the denominator. 91 is divisible by 7, and 91 divided by 7 is 13. 7 divided by 7 is 1. So, we now have (1 × 1 × 5 × 13) / (1 × 1 × 1 × 8). Now our multiplication is super manageable. Multiplying the numerators, we get 1 × 1 × 5 × 13 = 65. Multiplying the denominators, we get 1 × 1 × 1 × 8 = 8. So, our simplified fraction is 65/8. Simplifying before multiplying made our lives so much easier! We avoided dealing with huge numbers and made the calculation straightforward. Now, we have our answer in fraction form. The final step is to express it in its simplest form, which we’ll do in the next section. Great job so far! You’re doing awesome!

Step 3: Express the Result in Simplest Form

Okay, we've reached the final step! We've simplified the fractions, multiplied them together, and now we have R = 65/8. But we're not quite done yet. To truly solve the equation, we need to express this result in its simplest form. This means we need to determine if the fraction can be reduced further or if we should convert it to a mixed number. First, let’s check if 65/8 can be simplified. We need to see if 65 and 8 have any common factors other than 1. The factors of 65 are 1, 5, 13, and 65. The factors of 8 are 1, 2, 4, and 8. Looking at these lists, we can see that 65 and 8 do not share any common factors other than 1. This means that the fraction 65/8 is already in its simplest form as an improper fraction. However, in many cases, it’s more useful to express an improper fraction (where the numerator is greater than the denominator) as a mixed number. A mixed number is a whole number combined with a proper fraction (where the numerator is less than the denominator).

So, how do we convert 65/8 to a mixed number? We need to divide the numerator (65) by the denominator (8). When we divide 65 by 8, we get 8 as the quotient and 1 as the remainder. This tells us that 8 goes into 65 eight times with 1 left over. The quotient (8) becomes the whole number part of our mixed number, and the remainder (1) becomes the numerator of the fractional part. The denominator stays the same (8). So, 65/8 as a mixed number is 8 1/8. This means that R is equal to 8 and 1/8. Expressing the result in simplest form, whether as an improper fraction or a mixed number, is important because it gives us the clearest and most understandable answer. It’s like giving directions to someone; you want to provide them in the simplest and most direct way possible. In this case, 8 1/8 is a clear and concise way to express the value of R. And there we have it! We’ve successfully solved the equation R = (13/5) × (8/7) × (75/39) × (91/64) and found that R = 8 1/8. You guys did an amazing job following along step by step. This process of simplifying, multiplying, and expressing the result in its simplest form is a fundamental skill in math, and you’ve nailed it. Now you can confidently tackle similar problems. Great work!

Conclusion

So, we've reached the end of our mathematical journey, and what a journey it has been! We started with the equation R = (13/5) × (8/7) × (75/39) × (91/64) and, step by step, we broke it down, simplified it, and solved it. We discovered that R = 8 1/8. Remember, the key to solving complex math problems is to take them one step at a time. We first understood the equation, then we simplified the fractions, multiplied them together (simplifying along the way), and finally, we expressed our result in its simplest form. Each step was crucial, and together, they led us to the final answer. This process isn't just about getting the right answer; it’s about building a solid foundation in mathematical thinking. By learning to simplify fractions, multiply them efficiently, and express results clearly, you're developing skills that will help you in all areas of math and even in everyday life. Think about it – understanding fractions and proportions is essential in cooking, budgeting, measuring, and so much more!

I hope this step-by-step guide has made solving this equation clear and straightforward for you. Math doesn't have to be scary or overwhelming. When you break it down and tackle each part methodically, it becomes much more manageable—and even enjoyable! Keep practicing, keep asking questions, and keep exploring the world of mathematics. You've got this! And remember, every complex problem is just a series of smaller, simpler steps waiting to be solved. Thank you for joining me on this mathematical adventure. Keep up the fantastic work, and I look forward to exploring more math problems with you in the future. You're well on your way to becoming math whizzes!