Multiplying Fractions: Step-by-Step Examples & Solutions

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Hey guys! Today, we're diving into the world of multiplying fractions. It might seem tricky at first, but trust me, it's super straightforward once you get the hang of it. We're going to break down the process step-by-step and work through some examples together. So, grab your pencils and let's get started!

Understanding Fraction Multiplication

So, what's the big deal about multiplying fractions? The cool thing is, it's actually simpler than adding or subtracting them! The basic rule is this: to multiply fractions, you multiply the numerators (the top numbers) and then multiply the denominators (the bottom numbers). That's it!

Let's break that down a bit further. Think of a fraction as a part of a whole. When you multiply fractions, you're essentially finding a part of a part. For example, if you have half a pizza (1/2) and you want to eat a quarter (1/4) of that half, you're multiplying 1/4 by 1/2. The result will tell you what fraction of the whole pizza you're eating.

Why is this important? Well, fractions are everywhere! From cooking and baking to measuring and calculating, understanding how to multiply fractions is a crucial skill in everyday life. It's also a fundamental concept in math that will help you tackle more advanced topics later on. So, mastering this now will definitely pay off in the long run. We will solve together the following examples:

a) 3/4 * 1/2 b) 9/7 * 3/4 c) 8/5 * 7/8 d) 17/7 * 4/17 e) 2/3 * 1/4 * 8/5

Now, before we jump into specific examples, let's quickly recap the key steps:

  1. Multiply the numerators: Multiply the numbers on the top of the fractions.
  2. Multiply the denominators: Multiply the numbers on the bottom of the fractions.
  3. Simplify (if needed): Reduce the resulting fraction to its simplest form.

Got it? Great! Let's put these steps into action.

Example a) 3/4 * 1/2

Okay, let's start with our first example: 3/4 multiplied by 1/2. This is a classic example and a great way to understand the basic process.

Step 1: Multiply the numerators. The numerators are the top numbers in the fractions. In this case, we have 3 and 1. So, we multiply them together: 3 * 1 = 3.

Step 2: Multiply the denominators. The denominators are the bottom numbers. Here, we have 4 and 2. Multiplying them gives us: 4 * 2 = 8.

Step 3: Write the result. Now we combine the results from steps 1 and 2. The product of the numerators (3) becomes the new numerator, and the product of the denominators (8) becomes the new denominator. So, we have 3/8.

Step 4: Simplify (if needed). This is a crucial step! We need to check if our fraction, 3/8, can be simplified further. Simplifying means reducing the fraction to its lowest terms. To do this, we look for a common factor (a number that divides evenly into both the numerator and the denominator). In this case, 3 and 8 don't share any common factors other than 1. This means 3/8 is already in its simplest form.

Therefore, 3/4 * 1/2 = 3/8

See? Not so scary, right? We followed the steps, multiplied the numerators, multiplied the denominators, and simplified the result. Let's move on to the next example and keep practicing!

Example b) 9/7 * 3/4

Alright, let's tackle another one! This time we're multiplying 9/7 by 3/4. Remember the steps – we've got this!

Step 1: Multiply the numerators. We have 9 and 3 as our numerators. Multiplying them together: 9 * 3 = 27.

Step 2: Multiply the denominators. The denominators are 7 and 4. Let's multiply: 7 * 4 = 28.

Step 3: Write the result. Combining the results, we get 27/28.

Step 4: Simplify (if needed). Now, the crucial question: can we simplify 27/28? We need to find a common factor for both 27 and 28. Let's think... The factors of 27 are 1, 3, 9, and 27. The factors of 28 are 1, 2, 4, 7, 14, and 28. The only common factor is 1, which means 27/28 is already in its simplest form.

Therefore, 9/7 * 3/4 = 27/28

Great job! We're building momentum here. Notice how the process is the same, even with different numbers. We just follow the steps consistently, and we'll get to the right answer. Let's keep going!

Example c) 8/5 * 7/8

Okay, guys, let's keep the fraction multiplication train rolling! This time, we're tackling 8/5 multiplied by 7/8. This example has a little something extra that we'll need to pay attention to, so let's get to it!

Step 1: Multiply the numerators. Our numerators are 8 and 7. So, 8 * 7 = 56.

Step 2: Multiply the denominators. The denominators are 5 and 8. Multiplying them together: 5 * 8 = 40.

Step 3: Write the result. We now have the fraction 56/40.

Step 4: Simplify (if needed). Now, this is where things get interesting! Can we simplify 56/40? Absolutely! Both 56 and 40 are even numbers, which means they're both divisible by 2. But let's aim higher. Can we find a larger common factor? It turns out that both 56 and 40 are divisible by 8!

Let's divide both the numerator and the denominator by 8:

  • 56 / 8 = 7
  • 40 / 8 = 5

So, 56/40 simplifies to 7/5.

Therefore, 8/5 * 7/8 = 7/5

This example highlights the importance of simplifying fractions! Sometimes you can simplify along the way, and sometimes you need to do it at the end. Either way, always make sure your final answer is in its simplest form. 7/5 is an improper fraction (the numerator is greater than the denominator), and it can also be expressed as a mixed number: 1 2/5. But for this exercise, 7/5 is perfectly fine!

Example d) 17/7 * 4/17

Alright, team, let's keep crushing these fractions! Next up, we have 17/7 multiplied by 4/17. This one has a sneaky little trick in it, so keep your eyes peeled!

Step 1: Multiply the numerators. The numerators are 17 and 4. Multiplying them gives us: 17 * 4 = 68.

Step 2: Multiply the denominators. The denominators are 7 and 17. So, 7 * 17 = 119.

Step 3: Write the result. We now have the fraction 68/119.

Step 4: Simplify (if needed). Okay, this is where the trick comes in! 68 and 119 might seem like they don't have any common factors at first glance. But look closely... Do you notice anything about the original problem? We had 17 in both the numerator of the first fraction and the denominator of the second fraction! This means 17 is a common factor.

Let's divide both 68 and 119 by 17:

  • 68 / 17 = 4
  • 119 / 17 = 7

So, 68/119 simplifies to 4/7.

Therefore, 17/7 * 4/17 = 4/7

This example shows us that sometimes we can spot opportunities to simplify before we even multiply! If you see a common factor in the numerators and denominators, you can cancel them out to make the multiplication easier. This is called cross-cancelling, and it's a super useful shortcut.

Example e) 2/3 * 1/4 * 8/5

Okay, last one for today! This time, we're upping the ante with three fractions: 2/3 multiplied by 1/4 multiplied by 8/5. Don't worry, the same rules apply – we just have one extra step!

Step 1: Multiply the numerators. We have 2, 1, and 8 as our numerators. So, 2 * 1 * 8 = 16.

Step 2: Multiply the denominators. The denominators are 3, 4, and 5. Multiplying them together: 3 * 4 * 5 = 60.

Step 3: Write the result. Our fraction is now 16/60.

Step 4: Simplify (if needed). Time to simplify! Both 16 and 60 are even numbers, so we can definitely divide by 2. But let's see if we can go further. Both numbers are also divisible by 4!

Let's divide both the numerator and the denominator by 4:

  • 16 / 4 = 4
  • 60 / 4 = 15

So, 16/60 simplifies to 4/15.

Therefore, 2/3 * 1/4 * 8/5 = 4/15

Excellent work, everyone! We successfully multiplied three fractions together. The key takeaway here is that the process is the same, no matter how many fractions you're dealing with. Just multiply all the numerators, multiply all the denominators, and simplify at the end.

Key Takeaways for Multiplying Fractions

Okay, guys, we've covered a lot of ground today! Let's quickly recap the most important things to remember when multiplying fractions:

  • Multiply the numerators: Top numbers times top numbers.
  • Multiply the denominators: Bottom numbers times bottom numbers.
  • Simplify: Always reduce your answer to its simplest form. Look for common factors and divide both the numerator and denominator by them.
  • Cross-cancelling: If you see a common factor in the numerators and denominators before multiplying, you can cancel them out to make the problem easier.
  • Three or more fractions: The process is the same! Just multiply all the numerators and all the denominators.

Practice Makes Perfect

The best way to master multiplying fractions is to practice, practice, practice! Try working through some more examples on your own. You can find plenty of practice problems online or in math textbooks. Don't be afraid to make mistakes – that's how we learn! The more you practice, the more confident you'll become. And remember, if you get stuck, just go back to the steps we discussed and break the problem down. You've got this!

So, there you have it! Multiplying fractions demystified. I hope this guide has been helpful and that you're feeling more confident about tackling these problems. Keep up the great work, and I'll see you next time!