Math Problem Solutions: Fractions And Multiplication

Hey guys! Let's dive into solving some math problems involving fractions and multiplication. We've got three problems here that we're going to break down step by step. Math can seem tricky sometimes, but with a clear approach, we can conquer these challenges together. So, grab your pencils, and let’s get started!

Problem 1: 2.3/4 × 4

In this initial problem, we are tasked with multiplying a mixed number by a whole number: 2.3/4 × 4. To make this easier, let’s first convert the mixed number into an improper fraction. This involves multiplying the whole number part (2) by the denominator (4) and then adding the numerator (3). So, (2 * 4) + 3 = 11. We place this result over the original denominator, giving us 11/4. Now our problem looks like this: 11/4 × 4.

Next, we can rewrite the whole number 4 as a fraction by placing it over 1, making it 4/1. This doesn't change the value but helps us visualize the multiplication of fractions. Our problem now is: 11/4 × 4/1. To multiply fractions, we multiply the numerators together and the denominators together. So, 11 * 4 = 44, and 4 * 1 = 4. This gives us the fraction 44/4. Finally, we simplify the fraction 44/4 by dividing both the numerator and the denominator by their greatest common divisor, which is 4. 44 divided by 4 is 11, and 4 divided by 4 is 1. Therefore, the simplified fraction is 11/1, which is equal to 11. So, the solution to 2.3/4 × 4 is 11. Remember, converting mixed numbers to improper fractions and simplifying the final result are key steps in solving these types of problems. Practice these steps, and you’ll become a pro at fraction multiplication in no time!

Problem 2: 3.1/3 × 2/7

Let's tackle the second problem: 3.1/3 × 2/7. This one involves multiplying a mixed number by a proper fraction. Just like in the first problem, our first step is to convert the mixed number, 3.1/3, into an improper fraction. To do this, we multiply the whole number part (3) by the denominator (3) and then add the numerator (1). So, (3 * 3) + 1 = 10. We place this result over the original denominator, giving us 10/3. Now our problem looks like this: 10/3 × 2/7.

Now we can multiply the two fractions together. To multiply fractions, we multiply the numerators together and the denominators together. So, 10 * 2 = 20, and 3 * 7 = 21. This gives us the fraction 20/21. Next, we need to check if this fraction can be simplified. The factors of 20 are 1, 2, 4, 5, 10, and 20, while the factors of 21 are 1, 3, 7, and 21. The only common factor they share is 1, which means the fraction is already in its simplest form. Therefore, the solution to 3.1/3 × 2/7 is 20/21.

This problem highlights the importance of not only converting mixed numbers but also checking for simplification at the end. Simplifying fractions ensures that your answer is in the most reduced form, which is often the preferred way to express solutions. Keep practicing these steps, and you'll get more comfortable with multiplying and simplifying fractions!

Problem 3: 2.7/10 × 1.1/5

Okay, let’s move on to the third and final problem: 2.7/10 × 1.1/5. This problem, like the others, involves the multiplication of mixed numbers. Our first step remains the same: converting the mixed numbers into improper fractions. Let's start with 2.7/10. We multiply the whole number part (2) by the denominator (10) and then add the numerator (7). So, (2 * 10) + 7 = 27. We place this result over the original denominator, giving us 27/10. Now let's convert 1.1/5. We multiply the whole number part (1) by the denominator (5) and then add the numerator (1). So, (1 * 5) + 1 = 6. We place this result over the original denominator, giving us 6/5. Our problem now looks like this: 27/10 × 6/5.

Next, we multiply the two fractions together. Remember, we multiply the numerators together and the denominators together. So, 27 * 6 = 162, and 10 * 5 = 50. This gives us the fraction 162/50. Now, we need to simplify this fraction. Both 162 and 50 are even numbers, so they are both divisible by 2. Dividing 162 by 2 gives us 81, and dividing 50 by 2 gives us 25. So, our simplified fraction is 81/25. The factors of 81 are 1, 3, 9, 27, and 81, while the factors of 25 are 1, 5, and 25. Since they only share the factor 1, the fraction 81/25 is in its simplest form.

However, we can also express this improper fraction as a mixed number to get a better sense of its value. To do this, we divide 81 by 25. 25 goes into 81 three times (3 * 25 = 75), with a remainder of 6 (81 - 75 = 6). So, 81/25 is equal to 3 and 6/25. Therefore, the solution to 2.7/10 × 1.1/5 is 81/25 or 3 and 6/25.

This problem reinforces the importance of converting mixed numbers, multiplying fractions, simplifying the result, and even converting back to a mixed number if needed. Each of these steps helps us arrive at the most accurate and understandable solution. Keep up the great work, guys!

Key Takeaways

Alright, everyone, let’s recap what we’ve learned from these three problems. We've successfully navigated through multiplying mixed numbers and fractions, and now we have some solid steps to follow for future problems. First and foremost, we learned the importance of converting mixed numbers into improper fractions. This makes the multiplication process much smoother and prevents common errors. By converting, we ensure that we're working with a single numerator and denominator, which simplifies the multiplication.

Next, we focused on the actual multiplication process: multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator. This straightforward approach is the core of fraction multiplication. Once we have our new fraction, the job isn't quite done yet. Simplifying fractions is a crucial step. We looked for common factors between the numerator and denominator and divided both by their greatest common factor. This gives us the fraction in its simplest form, which is often required in math problems.

Finally, we touched on the idea of converting improper fractions back into mixed numbers. While not always necessary, it can provide a clearer understanding of the quantity, especially in real-world scenarios. Converting back allows us to express the answer in a way that might be more intuitive.

By mastering these steps—converting mixed numbers, multiplying fractions, simplifying, and converting back when necessary—you'll be well-equipped to tackle a wide range of fraction multiplication problems. Remember, practice makes perfect, so keep at it, and you'll become more confident with each problem you solve. Great job today, everyone!