Simplifying Fractions: Step-by-Step Solutions

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Fractions got you feeling frazzled? Don't worry, guys! We're about to break down how to simplify fractions like a pro. Whether you're tackling homework or just brushing up on your math skills, this guide will walk you through the process step-by-step. We'll take a look at several examples and simplify each one together. Let's dive in!

Understanding the Basics of Simplifying Fractions

Before we jump into specific problems, let's quickly recap what it means to simplify a fraction. At its core, simplifying a fraction means reducing it to its lowest terms. In simpler words, we want to find an equivalent fraction where the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. This makes the fraction as clean and concise as possible.

So, why is this important? Well, simplified fractions are easier to work with in calculations, and they give a clearer representation of the fraction's value. Imagine trying to compare 24/36 and 2/3 – the simplified form (2/3) instantly shows the fraction's value more clearly. The main keyword here is simplifying, which involves identifying and canceling common factors. Simplifying fractions is an essential skill in mathematics, ensuring numbers are in their most manageable form. Mastering this concept lays a strong foundation for more advanced mathematical operations, making calculations and problem-solving much more efficient. Let's get started and make fractions less intimidating!

Problem 1: (23 * 33 * 5) / (45 * 42 * 73 * 23)

Let's kick things off with our first fraction: (23 * 33 * 5) / (45 * 42 * 73 * 23). This looks a bit intimidating, but don't sweat it! We'll break it down step by step. The first thing to do is identify any common factors between the numerator (top part) and the denominator (bottom part). Remember, factors are numbers that divide evenly into another number.

In this case, we can see that both the numerator and the denominator have a factor of 23. So, let's cancel that out:

(23 * 33 * 5) / (45 * 42 * 73 * 23) becomes (33 * 5) / (45 * 42 * 73)

Now, let's look for more common factors. We can break down the numbers further:

  • 33 can be written as 3 * 11
  • 45 can be written as 3 * 15
  • 42 can be written as 6 * 7
  • 5 remains as 5
  • 73 remains as 73 (it's a prime number)

So, our fraction now looks like this: (3 * 11 * 5) / (3 * 15 * 6 * 7 * 73)

Aha! We see another common factor – 3. Let's cancel that out too:

(3 * 11 * 5) / (3 * 15 * 6 * 7 * 73) becomes (11 * 5) / (15 * 6 * 7 * 73)

Now, let's simplify further. We know that 15 is 3 * 5, so we can cancel out a factor of 5:

(11 * 5) / (15 * 6 * 7 * 73) becomes 11 / (3 * 6 * 7 * 73)

Multiplying out the denominator, we get:

11 / (3 * 6 * 7 * 73) = 11 / 9198

So, the simplified fraction is 11/9198. See? Not so scary when we take it one step at a time! The key to simplifying fractions lies in breaking down each number into its prime factors and canceling out the common ones. This methodical approach makes even the most complex fractions manageable.

Problem 2: (32 * 23) / (80 * 32)

Alright, let's move on to our second fraction: (32 * 23) / (80 * 32). At first glance, we can immediately spot a common factor – 32! This makes our job a whole lot easier. Canceling out the 32 in both the numerator and the denominator gives us:

(32 * 23) / (80 * 32) becomes 23 / 80

Now, we need to check if 23 and 80 have any other common factors. 23 is a prime number, meaning its only factors are 1 and itself. So, we just need to see if 23 divides evenly into 80.

Spoiler alert: it doesn't! If you try dividing 80 by 23, you'll get a decimal, not a whole number. This means that 23 and 80 share no common factors other than 1.

Therefore, the fraction 23/80 is already in its simplest form. That was quick, right? Sometimes simplifying fractions is as straightforward as spotting and canceling an obvious common factor. The key takeaway here is to always look for those immediate simplifications first; they can save you a lot of time and effort!

Problem 3: (16 * 49) / (11 * 9 * 5 * 52 * 33)

Now, let's tackle a slightly more complex fraction: (16 * 49) / (11 * 9 * 5 * 52 * 33). Don't be intimidated by the number of terms; we'll apply the same systematic approach of finding and canceling common factors.

First, let's break down each number into its prime factors:

  • 16 = 2 * 2 * 2 * 2 = 2^4
  • 49 = 7 * 7 = 7^2
  • 11 remains as 11 (prime number)
  • 9 = 3 * 3 = 3^2
  • 5 remains as 5 (prime number)
  • 52 = 2 * 2 * 13 = 2^2 * 13
  • 33 = 3 * 11

Now, our fraction looks like this: (2^4 * 7^2) / (11 * 3^2 * 5 * 2^2 * 13 * 3 * 11)

Let's start canceling out common factors. We see 2^4 in the numerator and 2^2 in the denominator. We can cancel out 2^2 from both, leaving us with 2^2 in the numerator:

(2^4 * 7^2) / (11 * 3^2 * 5 * 2^2 * 13 * 3 * 11) becomes (2^2 * 7^2) / (11 * 3^2 * 5 * 13 * 3 * 11)

Next, let's look for other common factors. We have 11 appearing twice in the denominator, but no 11 in the numerator. We also don't see any common factors between 7 and the remaining terms in the denominator.

So, let's multiply out the remaining terms:

  • Numerator: 2^2 * 7^2 = 4 * 49 = 196
  • Denominator: 11 * 3^2 * 5 * 13 * 3 * 11 = 11 * 9 * 5 * 13 * 3 * 11 = 21621

Our simplified fraction is now 196 / 21621. While this fraction still looks large, we've ensured that there are no common factors between the numerator and the denominator. Remember, simplifying fractions often involves breaking down complex numbers into their prime factors and systematically eliminating shared elements. This methodical approach ensures accuracy and helps to manage even the most daunting fractions.

Problem 4: (53 * 29) / (33 * 112)

Let's move on to our fourth fraction: (53 * 29) / (33 * 112). To simplify this, we'll follow our usual process of breaking down each number into its prime factors and then looking for common factors to cancel out.

Here's the breakdown:

  • 53 is a prime number, so it remains 53.
  • 29 is also a prime number, so it remains 29.
  • 33 can be factored into 3 * 11.
  • 112 can be factored into 2 * 2 * 2 * 2 * 7, or 2^4 * 7.

Now, let's rewrite the fraction with the prime factors:

(53 * 29) / (3 * 11 * 2^4 * 7)

Looking at the numerator and the denominator, we can see that there are no common factors. 53 and 29 are prime numbers and do not appear in the denominator's factors (3, 11, 2, and 7). This means that the fraction is already in its simplest form.

So, (53 * 29) / (33 * 112) is already simplified. Sometimes, fractions are presented in their simplest form right away, making our job much easier! In this case, the simplifying fractions process ends quickly because there are no shared factors to eliminate.

Problem 5: (53 * 29) / (33 * 112) - A Repeat?

Hey, wait a minute! This looks familiar... It's the same fraction as Problem 4: (53 * 29) / (33 * 112). We've already gone through the process of simplifying fractions for this one, and we found that it's already in its simplest form because there are no common factors between the numerator and the denominator.

So, the answer remains the same: (53 * 29) / (33 * 112) is already simplified.

Sometimes, you might encounter the same problem twice, especially in practice sets. It's a good reminder to double-check if you've solved it before, saving you time and effort!

Problem 6: (9 * 25) / (87 * 25)

Okay, let's tackle our final fraction: (9 * 25) / (87 * 25). We're back on track with a new problem, and this one has a clear common factor that jumps out right away – 25!

We can immediately cancel out the 25 in both the numerator and the denominator:

(9 * 25) / (87 * 25) becomes 9 / 87

Now, we need to check if 9 and 87 have any other common factors. Let's break them down:

  • 9 = 3 * 3 = 3^2
  • 87 = 3 * 29

Aha! We see a common factor of 3. Let's cancel that out:

9 / 87 becomes (3 * 3) / (3 * 29), which simplifies to 3 / 29

Now, 3 and 29 are both prime numbers, so they have no common factors other than 1. This means that 3/29 is the simplified form of our fraction.

So, the simplified fraction is 3/29. We started by spotting the obvious common factor (25), then broke down the remaining numbers to find another common factor (3). This systematic approach is the key to successful simplifying fractions.

Conclusion: Mastering the Art of Simplifying Fractions

Awesome job, guys! We've walked through several examples of simplifying fractions, and hopefully, you're feeling more confident about tackling these problems on your own. Remember, the key to simplifying fractions is to break down each number into its prime factors and then systematically cancel out any common factors between the numerator and the denominator.

Don't be afraid to take it one step at a time, and always double-check your work to make sure you haven't missed any common factors. With a little practice, you'll be simplifying fractions like a pro in no time! Keep up the great work, and happy calculating!