Solving Fractions: A Step-by-Step Guide

Hey Plastik Magazine readers! Let's dive into a cool math problem today. We're going to break down how to solve the expression: ${(3 \frac{1}{2}) * [\frac{2}{5} - (-\frac{7}{2}) * (-1 \frac{1}{5})]}$. Don't worry, it looks a bit scary at first, but we'll tackle it step by step. This is a great exercise to brush up on your fraction skills and remember the order of operations. Trust me, it's easier than it looks, and we'll have some fun along the way! So, grab your pencils and let's get started. By the end of this, you'll be a fraction-solving pro, ready to impress your friends (and maybe even yourself!).

Step 1: Convert Mixed Numbers to Improper Fractions

Alright, guys, our first step is to transform those pesky mixed numbers into something more manageable: improper fractions. Remember, a mixed number is a whole number and a fraction combined (like ${3 \frac{1}{2}}$), and an improper fraction is where the numerator (the top number) is bigger than the denominator (the bottom number). It's like changing your outfit; it's the same person, just a different style! For ${3 \frac{1}{2}}$, we multiply the whole number (3) by the denominator (2) and then add the numerator (1). So, ${3 * 2 + 1 = 7}$. We keep the same denominator, so ${3 \frac{1}{2}}$ becomes ${\frac{7}{2}}$. Easy peasy, right?

Next, let's tackle ${-1 \frac{1}{5}}$. Using the same logic, we multiply 1 by 5 and add 1, giving us 6. We keep the negative sign and the denominator, so ${-1 \frac{1}{5}}$ transforms into ${-\frac{6}{5}}$. Now our expression looks a little less intimidating, doesn't it? We've just simplified the look of the problem, and trust me, it's already a huge step forward. This conversion is crucial because it makes the following calculations much cleaner and less prone to errors. Think of it as preparing your ingredients before you start cooking – it sets you up for success. We are at the very beginning of the journey, but it is important to remember what we are doing every step of the way.

Now, the expression is: ${\frac{7}{2} * [\frac{2}{5} - (-\frac{7}{2}) * (-\frac{6}{5})]}$.

Why Convert to Improper Fractions?

You might be asking, why do we even need to do this? Well, converting to improper fractions simplifies multiplication and division. It makes it easier to keep track of the parts and ensures that you're working with a single, unified fraction. Mixed numbers can be tricky when you're multiplying or dividing, as you need to remember to multiply both the whole number and the fractional part. Improper fractions eliminate this extra step, allowing us to focus on the core calculation. Plus, it makes the entire problem more visually consistent and easier to follow.

Step 2: Multiply the Fractions Inside the Brackets

Alright, time to focus on the heart of the matter – the multiplication within those brackets! Remember, the order of operations (PEMDAS/BODMAS) tells us to handle things inside parentheses or brackets first. So, we're going to multiply ${-\frac{7}{2}}$ by ${-\frac{6}{5}}$. When multiplying fractions, we simply multiply the numerators together and the denominators together. So, ${-7 * -6 = 42}$ and ${2 * 5 = 10}$. Also, a negative times a negative equals a positive, so ${-\frac{7}{2} * -\frac{6}{5} = \frac{42}{10}}$.

Our expression now becomes: ${\frac{7}{2} * [\frac{2}{5} - \frac{42}{10}]}$. See, it's getting simpler with each step! We are going to continue going step by step.

Multiplying Fractions: A Refresher

Let's quickly recap the golden rule of fraction multiplication: Multiply the numerators and multiply the denominators. It's that simple! There's no need to find a common denominator (unlike addition and subtraction). Just straight multiplication. For example, if you had to multiply ${\frac{1}{2} * \frac{3}{4}}$, you'd do ${1 * 3 = 3}$ and ${2 * 4 = 8}$, giving you ${\frac{3}{8}}$. Keep this in mind, and you'll be a fraction multiplication master in no time.

Step 3: Subtract the Fractions Inside the Brackets

Now, we've got to subtract the fractions within the brackets: ${\frac{2}{5} - \frac{42}{10}}$. But wait! Before we subtract, we need to make sure our fractions have a common denominator. This is the crucial step for addition and subtraction. It's like comparing apples and oranges; you need to get them into the same unit before you can compare them. In this case, we can easily change ${\frac{2}{5}}$ to have a denominator of 10. We multiply both the numerator and the denominator by 2. So, ${\frac{2 * 2}{5 * 2} = \frac{4}{10}}$.

Now, the expression within the brackets is ${\frac{4}{10} - \frac{42}{10}}$. Subtracting the numerators, we get ${4 - 42 = -38}$. So, ${\frac{4}{10} - \frac{42}{10} = -\frac{38}{10}}$. We have made some real progress here! We are almost at the end.

Our expression is now: ${\frac{7}{2} * -\frac{38}{10}}$.

Finding a Common Denominator

Finding a common denominator is all about making the denominators the same. You need to identify the least common multiple (LCM) of the denominators. In our case, the LCM of 5 and 10 is 10. To get a denominator of 10, you multiply the numerator and denominator of the first fraction by a number that will result in the denominator being 10, while keeping the value of the fraction the same. Remember, whatever you do to the bottom (denominator), you must also do to the top (numerator). This step ensures you're comparing equal parts before you add or subtract.

Step 4: Multiply the Remaining Fractions

Alright, we're on the home stretch! We have ${\frac{7}{2} * -\frac{38}{10}}$. Multiply the numerators: ${7 * -38 = -266}$. Multiply the denominators: ${2 * 10 = 20}$. So, we have ${-\frac{266}{20}}$. Voila! We've done the main calculation!

Our final answer is ${-\frac{266}{20}}$. But wait, can we simplify this fraction? Yes, we can! Both the numerator and the denominator are divisible by 2. So, ${-\frac{266 \div 2}{20 \div 2} = -\frac{133}{10}}$.

Simplifying Fractions

Simplifying fractions is all about reducing them to their lowest terms. It's like trimming down a long word to its essential parts. You divide both the numerator and the denominator by their greatest common factor (GCF). In our example, the GCF of 266 and 20 was 2. Dividing both by 2 gave us the simplified form ${-\frac{133}{10}}$. This is equivalent to ${-13 \frac{3}{10}}$. Simplifying makes fractions easier to understand and work with, and it's always a good practice.

Step 5: Simplify and Convert to a Mixed Number (Optional)

Now we have ${-\frac{133}{10}}$. This is a perfectly valid answer, but for a nicer look, let's convert it back to a mixed number. How many times does 10 go into 133? It goes in 13 times with a remainder of 3. So, ${-\frac{133}{10}}$ is the same as ${-13 \frac{3}{10}}$. And there you have it, folks! We've successfully solved the expression. Great job, everyone!

Converting Improper Fractions to Mixed Numbers

To convert an improper fraction (like ${\frac{133}{10}}$) to a mixed number, you divide the numerator by the denominator. The whole number part of the mixed number is the quotient (the result of the division), the numerator of the fractional part is the remainder, and the denominator stays the same. So, when you divide 133 by 10, you get 13 with a remainder of 3, resulting in ${13 \frac{3}{10}}$. Remember to keep the negative sign if the original fraction was negative.

Conclusion

And that, my friends, is how you solve this math problem! We've gone from mixed numbers and brackets to a simplified mixed number. You've now conquered a tricky fraction problem, and you've reinforced your skills in order of operations, fraction multiplication, and fraction subtraction. Keep practicing, and you'll become a math whiz in no time. If you got lost somewhere, go back, review the steps, and try it again. You've got this!

This article provides a detailed step-by-step guide to solving the given mathematical expression. It covers converting mixed numbers to improper fractions, multiplying fractions, subtracting fractions, and simplifying the final answer. Each step is explained in a clear and easy-to-understand manner, making it accessible for readers of Plastik Magazine who may be looking to refresh their math skills or learn new concepts. The use of bold and italic text highlights key terms and concepts, while the conversational tone makes the learning process more engaging and less intimidating. The inclusion of additional tips and explanations further enhances understanding, ensuring that readers not only solve the problem but also grasp the underlying mathematical principles.