Simplifying And Solving Multiplication Problems

Hey guys! Let's dive into simplifying and solving some multiplication problems. This is a fundamental skill in math, and once you get the hang of it, you'll be breezing through these types of questions. We'll break down each problem step by step, so it's super clear how to arrive at the solution. Get ready to sharpen those math skills!

Problem a) $\frac{3}{5} \times \frac{1}{3} \times \frac{5}{4} =$

Okay, let's tackle the first multiplication problem: $\frac{3}{5} \times \frac{1}{3} \times \frac{5}{4}$. When we're multiplying fractions, the first thing we want to do is see if we can simplify anything before we actually multiply. This can save us a lot of work in the long run. Simplifying fractions involves looking for common factors between the numerators (the top numbers) and the denominators (the bottom numbers). If we spot any, we can divide both the numerator and the denominator by that common factor.

In this case, we can see that there's a 3 in the numerator of the first fraction and a 3 in the denominator of the second fraction. We can cancel these out, because $\frac{3}{3}$ equals 1. Similarly, there's a 5 in the numerator of the third fraction and a 5 in the denominator of the first fraction. These can also be canceled out for the same reason. So, after canceling, our expression looks like this:

$\frac{\cancel{3}}{5} \times \frac{1}{\cancel{3}} \times \frac{\cancel{5}}{4} = \frac{1}{1} \times \frac{1}{1} \times \frac{1}{4}$

Now, the multiplication becomes much simpler. We just multiply the numerators together (1 * 1 * 1 = 1) and the denominators together (1 * 1 * 4 = 4). This gives us the final answer:

$\frac{1}{4}$

So, $\frac{3}{5} \times \frac{1}{3} \times \frac{5}{4} = \frac{1}{4}$. See? Simplifying first can make things way easier!

Why Simplifying Fractions Matters

Simplifying fractions before multiplying isn't just a neat trick; it's a fundamental part of mathematical efficiency. Imagine if we hadn't simplified in this problem. We would have had to multiply $3 \times 1 \times 5$ to get 15 for the numerator, and $5 \times 3 \times 4$ to get 60 for the denominator. That would give us the fraction $\frac{15}{60}$. While this isn't wrong, we're not done yet. We would then have to simplify $\frac{15}{60}$ by finding the greatest common divisor (GCD) of 15 and 60, which is 15, and then dividing both the numerator and the denominator by 15. This gives us $\frac{1}{4}$, the same answer we got by simplifying upfront, but with more steps.

By simplifying first, we work with smaller numbers, reducing the chances of making errors and making the multiplication process smoother. This is especially helpful when you're dealing with larger numbers or more complex fractions.

Practicing Simplification

To get really good at simplifying fractions, practice is key. Look for opportunities to simplify in your math problems, and make it a habit. The more you do it, the quicker you'll become at spotting those common factors. Remember, the goal is to make the numbers as small as possible before you multiply, which will lead to less work and fewer opportunities for mistakes.

Also, it's worth noting that simplifying fractions ties into other areas of math, such as finding equivalent fractions and comparing fractions. The better you are at simplifying, the stronger your overall math foundation will be.

Problem c) $\frac{4}{6} \times \frac{8}{9} \times \frac{9}{3} =$

Alright, let’s move on to the next problem: $\frac{4}{6} \times \frac{8}{9} \times \frac{9}{3}$. Again, our first step should be to simplify before we multiply. This makes the numbers more manageable and reduces the risk of errors. Look closely at the fractions and see if there are any common factors we can cancel out between the numerators and denominators.

In this problem, we can see that there's a 9 in the numerator of the third fraction and a 9 in the denominator of the second fraction. Bingo! Let's cancel those out. Also, we can simplify $\frac{4}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This turns $\frac{4}{6}$ into $\frac{2}{3}$. Now, our expression looks like this:

$\frac{2}{3} \times \frac{8}{\cancel{9}} \times \frac{\cancel{9}}{3} = \frac{2}{3} \times \frac{8}{1} \times \frac{1}{3}$

Now we can multiply the numerators together: $2 \times 8 \times 1 = 16$. And we multiply the denominators together: $3 \times 1 \times 3 = 9$. This gives us the fraction $\frac{16}{9}$. This fraction is an improper fraction because the numerator is greater than the denominator. While $\frac{16}{9}$ is a perfectly valid answer, sometimes it's helpful to convert it into a mixed number. A mixed number has a whole number part and a fractional part.

To convert $\frac{16}{9}$ into a mixed number, we divide 16 by 9. 9 goes into 16 once, with a remainder of 7. So, the whole number part is 1, and the fractional part is $\frac{7}{9}$. Therefore, $\frac{16}{9}$ is equal to $1\frac{7}{9}$.

So, $\frac{4}{6} \times \frac{8}{9} \times \frac{9}{3} = \frac{16}{9} = 1\frac{7}{9}$. Great job!

Converting Improper Fractions to Mixed Numbers

Let's dive a bit deeper into why and how we convert improper fractions to mixed numbers. An improper fraction, as we mentioned, is a fraction where the numerator is greater than or equal to the denominator. Examples include $\frac{5}{3}$, $\frac{10}{4}$, and of course, our $\frac{16}{9}$. While improper fractions are mathematically sound, mixed numbers often give us a better sense of the size of the number.

Think of it this way: if you have $\frac{16}{9}$ of a pizza, it's not immediately clear how many whole pizzas you have and how much is left over. But if you know you have $1\frac{7}{9}$ pizzas, you instantly know you have one whole pizza and a little more.

The process of converting is straightforward: divide the numerator by the denominator. The quotient (the whole number result of the division) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part, and the denominator stays the same.

In our example, dividing 16 by 9 gives us a quotient of 1 and a remainder of 7. So, 1 is the whole number, 7 is the new numerator, and 9 remains the denominator. This gives us $1\frac{7}{9}$.

Why Mixed Numbers Matter

Mixed numbers are especially useful in real-world scenarios. For example, if you're measuring ingredients for a recipe, you might need $2\frac{1}{2}$ cups of flour. It's much more practical to use a mixed number than to say you need $\frac{5}{2}$ cups. Similarly, if you're measuring the length of something, you might get a result like $3\frac{3}{4}$ inches.

Understanding how to work with mixed numbers is also crucial for more advanced math topics, such as adding, subtracting, multiplying, and dividing mixed numbers. These operations often require converting mixed numbers back into improper fractions, performing the calculation, and then converting back to a mixed number if needed.

Problem d) $\frac{6}{12} \times \frac{4}{7} \times \frac{28}{16} =$

Last but not least, let's tackle this multiplication problem: $\frac{6}{12} \times \frac{4}{7} \times \frac{28}{16}$. By now, you know the drill: simplify first! Look for those common factors between the numerators and the denominators. Trust me, it makes everything smoother.

First, we can simplify $\frac{6}{12}$. Both 6 and 12 are divisible by 6, so we can divide both by 6 to get $\frac{1}{2}$. Next, let's look at $\frac{4}{7}$ and $\frac{28}{16}$. We can see that 7 goes into 28, so we can simplify these. 28 divided by 7 is 4, so we can think of $\frac{28}{7}$ as $\frac{4}{1}$ after canceling out the 7. Also, we have a 4 in the numerator of the second fraction and a 16 in the denominator of the third fraction. We can simplify these by dividing both by 4. This turns $\frac{4}{16}$ into $\frac{1}{4}$. Now, let's rewrite our expression with all the simplifications:

$\frac{1}{2} \times \frac{1}{1} \times \frac{4}{4}$

We can simplify $\frac{4}{4}$ to $\frac{1}{1}$. So now our equation is: $\frac{1}{2} \times \frac{1}{1} \times \frac{1}{1}$

Now, let's multiply the numerators together: $1 \times 1 \times 1 = 1$. And multiply the denominators together: $2 \times 1 \times 1 = 2$. This gives us the final answer:

$\frac{1}{2}$

So, $\frac{6}{12} \times \frac{4}{7} \times \frac{28}{16} = \frac{1}{2}$. Awesome!

Mastering Fraction Simplification

The key to mastering fraction simplification is to develop a keen eye for factors. The more comfortable you become with multiplication tables and divisibility rules, the easier it will be to spot opportunities for simplification. For instance, knowing that even numbers are always divisible by 2, or that numbers ending in 0 or 5 are divisible by 5, can be incredibly helpful.

Another strategy is to break down numbers into their prime factors. Prime factorization involves expressing a number as a product of its prime factors (numbers that are only divisible by 1 and themselves, like 2, 3, 5, 7, etc.). This can be particularly useful when dealing with larger numbers.

For example, let’s say you need to simplify the fraction $\frac{72}{96}$. You might not immediately see the greatest common factor, but if you break down 72 and 96 into their prime factors, the simplification becomes clearer:

• 72 = 2 x 2 x 2 x 3 x 3
• 96 = 2 x 2 x 2 x 2 x 2 x 3

Now you can easily see the common factors: three 2s and one 3. Multiplying these common factors gives you the greatest common factor (GCD), which is $2 \times 2 \times 2 \times 3 = 24$. Dividing both the numerator and the denominator by 24 simplifies the fraction to $\frac{3}{4}$.

Practice Makes Perfect

Like any math skill, simplifying fractions takes practice. The more you work with fractions, the more intuitive the process will become. Try challenging yourself with different types of problems, and don't be afraid to make mistakes. Mistakes are a valuable part of the learning process. They help you identify areas where you need more practice and refine your understanding.

And remember, guys, simplifying fractions isn't just a standalone skill. It's a foundational element that supports many other mathematical concepts. So, the time and effort you invest in mastering this skill will pay off in the long run. Keep practicing, stay curious, and you'll be simplifying fractions like a pro in no time!

Conclusion

So, there you have it! We've successfully simplified and solved three multiplication problems involving fractions. Remember, the key is to always look for opportunities to simplify before you multiply. This not only makes the calculations easier but also reduces the chances of making mistakes. Keep practicing, and you'll become a fraction-simplifying master in no time! You've got this!