Dividing Fractions: A Step-by-Step Guide To Simplest Form

Oct 11, 2025 by ADMIN 58 views

Hey guys! Let's dive into the world of dividing fractions and make sure we understand how to find that quotient and write it in its simplest form. Sounds fun, right? Don't worry, it's not as scary as it might seem at first. We'll break it down step by step, and by the end of this, you'll be a fraction-dividing pro. We're going to tackle a problem together: $7 rac{1}{2} rac{5}{6}$ . Our goal is to find the quotient, which is the result of a division problem, and express it as a simplified fraction. So, grab your pencils, and let's get started! This is more than just a math lesson; it's about building a solid foundation in fractions. Mastering this skill will boost your confidence in tackling more complex mathematical problems. We'll go through the concepts, step by step, with an example that we'll complete together, so you can be sure to follow along. Math might seem daunting at first. However, if you break down the steps, it will become much more accessible. We'll focus on clarity and accuracy, making sure you grasp each concept before moving on. Ready? Let's do this!

Converting Mixed Numbers to Improper Fractions

Alright, first things first! Before we can even think about dividing, we need to make sure we're working with fractions that are easy to manage. That's where converting mixed numbers to improper fractions comes into play. So, what's a mixed number and why do we care? A mixed number is a whole number and a fraction combined, like our $7 rac{1}{2}$ . An improper fraction, on the other hand, is when the numerator (the top number) is greater than the denominator (the bottom number). It's like saying you have more slices of pizza than the pizza was originally cut into! To change a mixed number to an improper fraction, we multiply the whole number by the denominator, then add the numerator, and keep the same denominator. Let's take our $7 rac{1}{2}$ and convert it. First, we multiply the whole number (7) by the denominator (2): $7 * 2 = 14$ . Then, we add the numerator (1): $14 + 1 = 15$ . Finally, we keep the same denominator, which is 2. So, $7 rac{1}{2}$ becomes $rac{15}{2}$ . See? Not so bad! This step simplifies the division process, making it much easier to handle. Now that we have our mixed number in improper form, we're ready to move to the next step of the process. This conversion is a crucial step because it allows us to treat all parts of the equation as fractions. By doing this, we ensure that all calculations are consistent, which is vital for getting the right answer. This is a common concept throughout mathematics, particularly when dealing with fractions.

Example: Converting $7 rac{1}{2}$ to an improper fraction

Let's go through the conversion of $7 rac{1}{2}$ again, step by step, to make sure we've got it. This is a crucial step in our division process. Without it, we'd be stuck trying to divide a mixed number by a fraction, which is more difficult. Let's break it down further:

Multiply the whole number by the denominator: We start with the whole number, which is 7, and multiply it by the denominator of the fraction, which is 2. So, $7 * 2 = 14$ .
Add the numerator: Next, we take the result from the previous step (14) and add the numerator of the fraction, which is 1. So, $14 + 1 = 15$ .
Keep the same denominator: Finally, we keep the same denominator as the original fraction, which is 2. So, our new fraction becomes $rac{15}{2}$ .

Therefore, $7 rac{1}{2}$ converted to an improper fraction is $rac{15}{2}$ .

Dividing Fractions: The "Keep, Change, Flip" Method

Now for the fun part: dividing fractions! This is where the "Keep, Change, Flip" method comes in handy. It's a super simple trick that makes dividing fractions a breeze. Here's how it works: First, you keep the first fraction as it is. Then, you change the division sign to a multiplication sign. Finally, you flip (or find the reciprocal of) the second fraction. Let's apply this to our problem: We had $rac{15}{2} rac{5}{6}$ . We keep $rac{15}{2}$ . We change the division to multiplication, so now we have a multiplication problem. We flip $rac{5}{6}$ to become $rac{6}{5}$ . Now, our problem looks like this: $rac{15}{2} * rac{6}{5}$ . See, much easier to manage! This method is a mathematical shortcut that is extremely powerful. It transforms a division problem into a multiplication problem. The conversion is key to simplifying the process and achieving accurate results. This method is efficient and can be used to divide fractions quickly. Mastering the "Keep, Change, Flip" method is essential for fraction division. Let's go through the example to clear any questions you may have on this step!

Applying "Keep, Change, Flip" to our Problem

Okay, let's get down to the nitty-gritty and apply the "Keep, Change, Flip" method to our problem step-by-step to make sure we understand it well. Remember, our original problem was $7 rac{1}{2} rac{5}{6}$ , and after converting to an improper fraction, it became $rac{15}{2} rac{5}{6}$ . Here’s how we apply the rule:

Keep the first fraction: The first fraction in our new problem is $rac{15}{2}$ . We simply keep this fraction as it is. So, we still have $rac{15}{2}$ .
Change the division sign: The original operation was division ( $rac{15}{2} rac{5}{6}$ ). We change the division sign to a multiplication sign. So, now our problem is $rac{15}{2} * rac{5}{6}$ .
Flip the second fraction: The second fraction is $rac{5}{6}$ . We flip it to find its reciprocal. This means we swap the numerator and the denominator, making it $rac{6}{5}$ .

Now our problem looks like this: $rac{15}{2} * rac{6}{5}$ . We have successfully transformed the division problem into a multiplication problem, and we're ready to solve it!

Multiplying the Fractions

Now that we have our fractions set up as a multiplication problem, let's multiply! Multiplying fractions is straightforward. You multiply the numerators together to get the new numerator, and you multiply the denominators together to get the new denominator. For our problem, we have $rac{15}{2} * rac{6}{5}$ . Multiply the numerators: $15 * 6 = 90$ . Multiply the denominators: $2 * 5 = 10$ . So, our answer before simplifying is $rac{90}{10}$ . Easy peasy, right? Remember, when you multiply fractions, you're essentially finding a part of a part. The multiplication process combines the two fractions into one. This step is a fundamental skill in fraction arithmetic. It's key to moving to the final step and achieving the final answer. Let's practice this step using our example to ensure it's clear to understand.

Multiplying the Numerators and Denominators

Let's get into how to multiply the fractions, going over the actual process. In our multiplication problem, we have $rac{15}{2} * rac{6}{5}$ . Here’s how we do it:

Multiply the numerators: The numerators are the top numbers of the fractions. In our case, these are 15 and 6. Multiplying them gives us: $15 * 6 = 90$ . This is the numerator of our new fraction.
Multiply the denominators: The denominators are the bottom numbers of the fractions. In our case, these are 2 and 5. Multiplying them gives us: $2 * 5 = 10$ . This is the denominator of our new fraction.

So, after multiplying, we get $rac{90}{10}$ . We're almost there! Now we have one final step to wrap up the process.

Simplifying the Fraction

Finally, we need to simplify our fraction to its simplest form. This means reducing it as much as possible. To do this, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both the numerator and denominator evenly. In our case, our fraction is $rac{90}{10}$ . Both 90 and 10 are divisible by 10. So, we divide both the numerator and the denominator by 10. $90 rac{10 = 9$ and $10 rac{10 = 1$ . Therefore, our simplified fraction is $rac{9}{1}$ , which simplifies to just 9. And there you have it, guys! The quotient of $7 rac{1}{2} rac{5}{6}$ is 9. Always remember to simplify your fractions to the simplest form, as it's the standard way to express the answer. This final step makes our answer more manageable. It also helps us understand the true relationship between the numerator and the denominator. Let's take a closer look at this simplification to solidify our understanding.

Simplifying $rac{90}{10}$ to Its Simplest Form

Let's simplify $rac{90}{10}$ to its simplest form. The goal here is to reduce the fraction until the numerator and denominator have no common factors other than 1. Here’s the process:

Find the greatest common divisor (GCD): The GCD is the largest number that divides both 90 and 10 evenly. In this case, the GCD is 10.
Divide the numerator and denominator by the GCD: Divide both the numerator (90) and the denominator (10) by the GCD (10).
- $90 rac{10 = 9$
- $10 rac{10 = 1$
Write the simplified fraction: The simplified fraction is $rac{9}{1}$ . Since any number divided by 1 is itself, we can simplify this further to just 9.

So, the simplified form of $rac{90}{10}$ is 9. That’s how we arrived at our final answer.

Conclusion

Awesome job, everyone! We've successfully divided fractions and written the answer in its simplest form! We converted the mixed number to an improper fraction, used the "Keep, Change, Flip" method, multiplied the fractions, and simplified the result. Remember, practice makes perfect. The more you work with fractions, the more comfortable you'll become. Keep at it, and you'll master these concepts in no time. Keep practicing, and you'll get better with each problem you solve. You're now well-equipped to handle fraction division. Keep up the great work, and don't hesitate to revisit these steps whenever you need a refresher. You've got this!