Dividing Fractions And Mixed Numbers Step-by-Step

Hey guys! Let's break down these division problems step by step. We'll tackle fractions and mixed numbers, making sure you understand each part of the process. Think of it like this: dividing by a fraction is the same as multiplying by its inverse (flipping it over). Ready to dive in?

Understanding Division with Fractions

Before we get started, let’s recap the basic concept of dividing fractions. The key thing to remember is that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply the fraction flipped over. For example, the reciprocal of 1/2 is 2/1 (which is just 2). This might seem a bit confusing at first, but it’s a super useful trick that makes fraction division way easier. When you divide by a fraction, you're essentially asking, "How many of this fraction fit into the number I'm dividing?" Multiplying by the reciprocal helps us answer that question directly. This concept is crucial not only for solving these problems but also for understanding more complex math topics later on. So, let’s keep this reciprocal trick in our mental toolkit as we go through the examples. By understanding the concept behind flipping the fraction and multiplying, you'll gain a deeper grasp of how division works with fractions. Now, let's move on to the first problem and see how this works in practice!

Problem 1: ${ \frac{3}{8} \div \frac{1}{2} }$

Okay, let's tackle the first problem: ${ \frac{3}{8} \div \frac{1}{2} }$. The first step, as we talked about, is to change the division to multiplication by flipping the second fraction (the divisor) to its reciprocal. So, ${ \frac{1}{2} }$ becomes ${ \frac{2}{1} }$, which is just 2. Now our problem looks like this: ${ \frac{3}{8} \times \frac{2}{1} }$. To multiply fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, we have (3 * 2) / (8 * 1), which equals 6/8. But wait, we're not done yet! We need to simplify this fraction. Both 6 and 8 can be divided by 2. When we divide both the numerator and the denominator by 2, we get ${ \frac{3}{4} }$. Ta-da! The answer to ${ \frac{3}{8} \div \frac{1}{2} }$ is ${ \frac{3}{4} }$. See? Not so scary when we break it down step by step. Remember, the key is to flip the second fraction and multiply, then simplify if you can. Now you've got the basic idea down, let’s move on to the next challenge. We'll build on this knowledge and tackle mixed numbers next, which might seem a bit more complicated, but we'll handle it together!

Problem 2: ${ 3 \frac{5}{9} \div \frac{1}{3} }$

Now let's dive into the second problem: ${ 3 \frac{5}{9} \div \frac{1}{3} }$. This one looks a little trickier because we've got a mixed number in the mix. No sweat, guys! The first thing we need to do is convert that mixed number into an improper fraction. Remember, a mixed number has a whole number part and a fraction part. To convert it, we multiply the whole number (3) by the denominator of the fraction (9), and then add the numerator (5). This gives us (3 * 9) + 5 = 27 + 5 = 32. We put this result over the original denominator, so ${ 3 \frac{5}{9} }$ becomes ${ \frac{32}{9} }$. Now our problem looks like this: ${ \frac{32}{9} \div \frac{1}{3} }$. Next, we apply our trusty rule: dividing by a fraction is the same as multiplying by its reciprocal. So we flip ${ \frac{1}{3} }$ to ${ \frac{3}{1} }$ and change the division to multiplication: ${ \frac{32}{9} \times \frac{3}{1} }$. Multiply the numerators: 32 * 3 = 96. Multiply the denominators: 9 * 1 = 9. This gives us ${ \frac{96}{9} }$. Time to simplify! Both 96 and 9 are divisible by 3. Dividing both by 3, we get ${ \frac{32}{3} }$. We can also convert this improper fraction back into a mixed number. 32 divided by 3 is 10 with a remainder of 2, so our final answer is ${ 10 \frac{2}{3} }$. See? We handled that mixed number like pros! By converting to an improper fraction first, we made the division much more manageable. Keep this technique in mind, as it's super helpful for any problem involving mixed numbers and fractions.

Problem 3: ${ \frac{2}{5} \div 1 \frac{1}{5} }$

Alright, let's tackle our third and final problem: ${ \frac{2}{5} \div 1 \frac{1}{5} }$. Just like in the last one, we've got a mixed number, so our first step is to convert it into an improper fraction. We'll take ${ 1 \frac{1}{5} }$ and multiply the whole number (1) by the denominator (5), then add the numerator (1). This gives us (1 * 5) + 1 = 5 + 1 = 6. Put that over the original denominator, and we get ${ \frac{6}{5} }$. Now our problem looks like this: ${ \frac{2}{5} \div \frac{6}{5} }$. Time for our favorite trick: flip the second fraction and multiply! We change ${ \frac{6}{5} }$ to ${ \frac{5}{6} }$ and the division becomes multiplication: ${ \frac{2}{5} \times \frac{5}{6} }$. Multiply the numerators: 2 * 5 = 10. Multiply the denominators: 5 * 6 = 30. We have ${ \frac{10}{30} }$. Now we simplify. Both 10 and 30 can be divided by 10, which gives us ${ \frac{1}{3} }$. And there we have it! ${ \frac{2}{5} \div 1 \frac{1}{5} }$ equals ${ \frac{1}{3} }$. You’ve nailed it! By consistently following our steps – converting mixed numbers to improper fractions, flipping the second fraction, multiplying, and simplifying – you can conquer any division problem with fractions. This approach not only gets you the right answer but also helps build a solid understanding of the underlying math principles.

Wrapping It Up

So, there you have it! We've solved three different division problems involving fractions and mixed numbers. Remember, the key takeaways are:

• Dividing by a fraction is the same as multiplying by its reciprocal (flipping it).
• Convert mixed numbers to improper fractions before dividing.
• Always simplify your answer if possible.

With these steps in mind, you'll be able to confidently tackle any fraction division problem that comes your way. Keep practicing, and you'll become a fraction division pro in no time! You've got this, and math can be fun when you break it down step by step. Don't hesitate to revisit these steps whenever you need a refresher, and remember that each problem you solve builds your skills and confidence. Keep up the great work, and you'll find that even the trickiest math challenges can be overcome with a little bit of practice and the right approach.