Multiplying Fractions: Step-by-Step Solutions

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Hey guys! Let's dive into the world of fraction multiplication. It might seem tricky at first, but I promise, once you get the hang of it, it's super straightforward. We're going to tackle some examples step-by-step, so you can see exactly how it's done. Let's get started!

Understanding Fraction Multiplication

Before we jump into specific problems, let's quickly recap the basics. Multiplying fractions is different from adding or subtracting them. The key thing to remember is that when you multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. That’s it! If you have mixed numbers, the first step is to convert them into improper fractions.

So, the general rule looks like this:

(a/b) * (c/d) = (a * c) / (b * d)

Where 'a' and 'c' are the numerators, and 'b' and 'd' are the denominators. Now that we've refreshed the basics, let's apply this to some real examples. We'll go through each one, showing every step, so you can follow along easily. Ready? Let's do this!

Solving the Problems

Now, let's break down each of the fraction multiplication problems provided. We'll go through each one step-by-step, so you can clearly see how to arrive at the solution. Remember, the key is to multiply the numerators together and the denominators together. And don't forget to simplify your answer if possible!

a) 9/10 * 5/6

Okay, let's start with our first problem: 9/10 multiplied by 5/6. The first step, as we discussed, is to multiply the numerators and the denominators separately.

So, we have:

(9 * 5) / (10 * 6)

This gives us:

45 / 60

Now, we need to simplify this fraction. Both 45 and 60 are divisible by 15, so we can divide both the numerator and the denominator by 15:

(45 ÷ 15) / (60 ÷ 15) = 3/4

Therefore, 9/10 * 5/6 = 3/4. We took our original multiplication problem, multiplied straight across, and then simplified to get our final answer. Remember, simplifying fractions is crucial to get the most reduced form. This makes the answer cleaner and easier to work with if you need to use it in further calculations.

b) 6/25 * 20/21

Next up, we have 6/25 multiplied by 20/21. We're going to use the same principle here: multiply the numerators and denominators first, and then simplify the result.

So, let's multiply:

(6 * 20) / (25 * 21)

This gives us:

120 / 525

This looks like a big fraction, but don't worry, we can simplify it. Both 120 and 525 are divisible by 15. So, let's divide both by 15:

(120 ÷ 15) / (525 ÷ 15) = 8/35

So, 6/25 * 20/21 simplifies to 8/35. You see, even with seemingly large numbers, simplifying can lead us to a much cleaner and understandable fraction. It's all about finding the greatest common divisor (GCD) and dividing both parts of the fraction by it.

c) 17/30 * 26/51

Let's tackle the third problem: 17/30 multiplied by 26/51. This one might look a bit intimidating, but we'll break it down just like the others. Our first step is always the same – multiply those numerators and denominators!

So, we set it up like this:

(17 * 26) / (30 * 51)

Multiplying these numbers gives us:

442 / 1530

Okay, now we need to simplify this fraction. Both 442 and 1530 are even numbers, so we know they're at least divisible by 2. Let's start there:

(442 ÷ 2) / (1530 ÷ 2) = 221 / 765

Now, this is still a bit clunky. It turns out that both 221 and 765 are divisible by 17. So, let's divide by 17:

(221 ÷ 17) / (765 ÷ 17) = 13/45

Therefore, 17/30 * 26/51 simplifies to 13/45. It sometimes takes a couple of steps to fully simplify a fraction, but that's perfectly normal. Just keep looking for common factors until you can't simplify any further.

d) 40/7 * 14/5

Moving on to our fourth problem, we've got 40/7 multiplied by 14/5. Let’s keep the momentum going by multiplying across – numerators times numerators, denominators times denominators.

We write it out like this:

(40 * 14) / (7 * 5)

Doing the multiplication, we get:

560 / 35

This fraction definitely looks like it can be simplified. Both 560 and 35 are divisible by 5, so let’s start there:

(560 ÷ 5) / (35 ÷ 5) = 112 / 7

Now, we can simplify even further! We see that 112 is divisible by 7. So, let’s divide:

(112 ÷ 7) / (7 ÷ 7) = 16/1 = 16

So, 40/7 * 14/5 simplifies all the way down to 16. This is a great example of how simplifying can really make a difference, turning a fraction into a whole number. Always keep an eye out for opportunities to simplify, it makes the final result much cleaner.

e) 57/37 * 74/86

Alright, let’s dive into the fifth problem: 57/37 multiplied by 74/86. As always, we start by multiplying the numerators and the denominators.

This gives us:

(57 * 74) / (37 * 86)

Performing the multiplication, we get:

4218 / 3182

These are some big numbers, but don't let that scare you! We can simplify this. Both numbers are even, so let's start by dividing by 2:

(4218 ÷ 2) / (3182 ÷ 2) = 2109 / 1591

Now, let's look for more common factors. It turns out that both 2109 and 1591 are divisible by 37. So, let’s divide:

(2109 ÷ 37) / (1591 ÷ 37) = 57/43

Thus, 57/37 * 74/86 simplifies to 57/43. This one required a couple of steps of simplification, but we got there in the end. The key is to keep breaking the numbers down until you can't find any more common factors.

f) 81/115 * 23/54

Last but not least, we have the sixth problem: 81/115 multiplied by 23/54. We're going to finish strong by applying the same method we've used throughout: multiply the numerators and the denominators.

So, we set it up as:

(81 * 23) / (115 * 54)

Multiplying these, we get:

1863 / 6210

Now for the simplification. Both 1863 and 6210 are divisible by 9, so let’s divide by 9:

(1863 ÷ 9) / (6210 ÷ 9) = 207 / 690

We can simplify further! Both 207 and 690 are divisible by 23. Dividing both by 23, we get:

(207 ÷ 23) / (690 ÷ 23) = 9/30

One more step! Both 9 and 30 are divisible by 3:

(9 ÷ 3) / (30 ÷ 3) = 3/10

Therefore, 81/115 * 23/54 simplifies to 3/10. That was a multi-step simplification, but we got there! It just goes to show that sometimes you need to keep simplifying until you reach the simplest form.

Key Takeaways

Okay, guys, we've tackled quite a few fraction multiplication problems, and I hope you're feeling more confident about it now. Let’s recap the main points to keep in mind:

  1. Multiply Numerators and Denominators: This is the fundamental step. Multiply the top numbers (numerators) together and the bottom numbers (denominators) together.
  2. Simplify, Simplify, Simplify: Always look for opportunities to simplify your fractions. Divide both the numerator and the denominator by their greatest common factor. This may take a few steps, but it’s worth it.
  3. Look for Common Factors: Identifying common factors between the numerator and denominator is key to simplifying efficiently.
  4. Practice Makes Perfect: The more you practice, the quicker you'll become at spotting common factors and simplifying fractions.

Practice Problems

To really solidify your understanding, try these practice problems. Work through them using the steps we’ve discussed, and remember to simplify your answers!

  1. 4/9 * 3/8
  2. 15/28 * 14/25
  3. 21/34 * 17/49

Work these out, and you’ll be a fraction multiplication pro in no time. Good luck, and keep up the great work!

Conclusion

So, guys, we've walked through a bunch of examples of multiplying fractions, and hopefully, you feel a lot more comfortable with the process. Remember, the key is to multiply straight across and then simplify. It's like a little puzzle – finding the common factors and reducing the fraction to its simplest form. Don't get discouraged if it takes a few steps to simplify; just keep going until you can't reduce it any further. Keep practicing, and you'll get the hang of it in no time. Whether you're working on homework, helping a friend, or just brushing up on your math skills, you've now got the tools to tackle fraction multiplication with confidence. You've got this!