Multiplying Fractions: Simplify 7/6 X 2
Hey guys! Let's dive into multiplying fractions, specifically tackling the problem of simplifying 7/6 multiplied by 2. It might seem a bit tricky at first, but trust me, it's totally manageable once we break it down step by step. We’ll go through the process together, making sure you understand not just how to do it, but also why it works. So, grab your pencils and let’s get started!
Understanding the Basics of Fraction Multiplication
Before we jump into the specifics of 7/6 multiplied by 2, let’s quickly recap the fundamentals of fraction multiplication. When you're multiplying fractions, the basic rule is pretty straightforward: you multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together.
But what happens when you're multiplying a fraction by a whole number, like in our case? Well, the trick is to remember that any whole number can be written as a fraction by simply putting it over a denominator of 1. For example, the whole number 2 can be expressed as the fraction 2/1. This makes the multiplication process consistent and easier to visualize.
To really solidify this, let’s walk through a simpler example. Imagine we want to multiply 1/2 by 3. First, we rewrite 3 as 3/1. Then, we multiply the numerators (1 x 3 = 3) and the denominators (2 x 1 = 2). So, 1/2 multiplied by 3 equals 3/2. See? Not too scary, right? Understanding this foundational concept is crucial because it's the key to tackling more complex problems like our main question: simplifying 7/6 multiplied by 2.
Now, let’s think about why this method works. When we multiply fractions, we are essentially finding a fraction of a fraction or, in this case, scaling a fraction by a whole number. Multiplying the numerators tells us how many parts we now have, and multiplying the denominators tells us the size of those parts relative to the whole. By understanding this, you're not just memorizing a rule; you're grasping the underlying logic, which will help you tackle all sorts of fraction problems with confidence.
Remember this key takeaway: To multiply a fraction by a whole number, rewrite the whole number as a fraction with a denominator of 1, then multiply the numerators and the denominators. This simple step makes the process consistent and much easier to handle. Keep this in mind as we move on to our main problem, and you’ll see how smoothly it applies!
Step-by-Step: Multiplying 7/6 by 2
Okay, let's get down to business and tackle the problem at hand: multiplying the fraction 7/6 by the whole number 2. We're going to break it down into simple, easy-to-follow steps, so you can see exactly how it's done.
Step 1: Rewrite the Whole Number as a Fraction
The first thing we need to do, as we discussed earlier, is to rewrite the whole number 2 as a fraction. To do this, we simply put it over a denominator of 1. So, 2 becomes 2/1. Now our problem looks like this:
7/6 * 2/1
This might seem like a small change, but it's a crucial step because it allows us to apply the standard rule for fraction multiplication. It makes the whole process much clearer and easier to manage. Plus, it sets the stage for the next step.
Step 2: Multiply the Numerators
Next up, we multiply the numerators. The numerator of the first fraction is 7, and the numerator of the second fraction is 2. So, we multiply these together:
7 * 2 = 14
This gives us the new numerator for our answer. We're halfway there! Multiplying the numerators is all about figuring out how many parts we have in total after the multiplication. In this case, we've got 14 parts, but we still need to determine the size of those parts relative to the whole.
Step 3: Multiply the Denominators
Now, let's multiply the denominators. The denominator of the first fraction is 6, and the denominator of the second fraction is 1. So, we multiply these together:
6 * 1 = 6
This gives us the new denominator for our answer. Remember, the denominator tells us how many parts make up a whole. In this case, each part is 1/6 of the whole. So, after multiplying the denominators, we know we're dealing with sixths.
Step 4: Combine the New Numerator and Denominator
We've now got our new numerator (14) and our new denominator (6). We combine these to form our new fraction:
14/6
This is the result of multiplying 7/6 by 2. However, we're not quite done yet. The final step is crucial: we need to simplify this fraction to its simplest form. Simplifying fractions makes them easier to understand and work with, so it's a skill worth mastering.
By following these steps, you can confidently multiply any fraction by a whole number. Rewrite the whole number as a fraction, multiply the numerators, multiply the denominators, and then combine the results. But remember, the journey doesn't end there. The final and equally important step is simplification, which we'll dive into next!
Simplifying the Fraction 14/6
Alright, we've multiplied 7/6 by 2 and arrived at the fraction 14/6. That’s a solid first step, but in math, we always want to express our answers in the simplest form possible. Simplifying fractions makes them easier to understand and use in further calculations. So, how do we simplify 14/6? Let's break it down.
Step 1: Find the Greatest Common Factor (GCF)
The first thing we need to do is find the greatest common factor (GCF) of the numerator (14) and the denominator (6). The GCF is the largest number that divides evenly into both numbers. To find it, we can list the factors of each number:
- Factors of 14: 1, 2, 7, 14
- Factors of 6: 1, 2, 3, 6
Looking at these lists, we can see that the greatest common factor of 14 and 6 is 2. This means that 2 is the largest number that can divide both 14 and 6 without leaving a remainder. Finding the GCF is a crucial step because it tells us the largest amount we can reduce the fraction by in one go.
Step 2: Divide Both Numerator and Denominator by the GCF
Now that we've found the GCF, we divide both the numerator and the denominator by it. This is the key to simplifying the fraction. We're essentially reducing the fraction to its lowest terms while maintaining its value.
So, we divide 14 by 2:
14 ÷ 2 = 7
And we divide 6 by 2:
6 ÷ 2 = 3
By dividing both the top and bottom by the same number, we're ensuring that we're not changing the fraction's value, just its appearance. It’s like cutting a pizza into fewer, but larger, slices – you still have the same amount of pizza!
Step 3: Write the Simplified Fraction
After dividing by the GCF, we get our simplified numerator and denominator. So, our simplified fraction is:
7/3
This fraction is in its simplest form because 7 and 3 have no common factors other than 1. We've successfully reduced 14/6 to 7/3, which is much cleaner and easier to work with. But wait, there's one more thing we can do to make this answer even clearer!
Simplifying fractions is a fundamental skill in mathematics, and it's essential for clear communication and efficient problem-solving. By finding the GCF and dividing, we've made our fraction as simple as possible. But let's take it a step further and convert this improper fraction into a mixed number for an even better understanding of its value.
Converting 7/3 to a Mixed Number
Okay, so we've simplified our fraction to 7/3, which is fantastic! But you might notice that 7/3 is an improper fraction. This means that the numerator (7) is larger than the denominator (3). While 7/3 is a perfectly correct answer, it's often helpful to convert improper fractions into mixed numbers. Mixed numbers give us a clearer sense of the actual value because they show the whole number part and the remaining fractional part separately.
So, how do we convert 7/3 into a mixed number? Don't worry; it's a straightforward process. Let's break it down:
Step 1: Divide the Numerator by the Denominator
The first step is to divide the numerator (7) by the denominator (3). We're trying to figure out how many whole times 3 goes into 7. So, we perform the division:
7 ÷ 3 = 2 with a remainder of 1
This tells us that 3 goes into 7 two whole times, and we have 1 left over. This is the key information we need to form our mixed number. The whole number part comes from the number of times the denominator goes into the numerator, and the remainder will form the fractional part.
Step 2: Write the Whole Number Part
From the division, we found that 3 goes into 7 two whole times. So, the whole number part of our mixed number is 2. We've now got the big, main number that represents the complete wholes in our fraction.
Step 3: Form the Fractional Part
Next, we need to form the fractional part of our mixed number. The remainder from our division (which was 1) becomes the new numerator, and the original denominator (3) stays the same. So, the fractional part is 1/3.
Remember, the fractional part represents the portion that's less than a whole. In this case, we have one-third of a whole left over after taking out the whole numbers.
Step 4: Combine the Whole Number and the Fractional Part
Finally, we combine the whole number part (2) and the fractional part (1/3) to form our mixed number:
2 1/3
So, the improper fraction 7/3 is equivalent to the mixed number 2 1/3. This means that 7/3 is equal to two whole units and one-third of another unit. This representation often makes it easier to visualize and understand the quantity we're dealing with.
Converting improper fractions to mixed numbers is a valuable skill because it provides a more intuitive understanding of the fraction's value. It's especially useful when dealing with real-world situations, like measuring ingredients for a recipe or figuring out how much pizza each person gets. So, remember this process: divide, write the whole number, form the fraction, and combine. You'll be a pro at mixed numbers in no time!
Final Answer: 7/6 Multiplied by 2 in Simplest Form
Alright, let's bring it all together! We started with the problem of multiplying 7/6 by 2 and expressing the answer as a fraction in simplest form. We've gone through all the steps, and now we're ready to state our final answer.
First, we rewrote the whole number 2 as a fraction, making it 2/1. Then, we multiplied the numerators (7 * 2 = 14) and the denominators (6 * 1 = 6) to get the fraction 14/6. After that, we simplified 14/6 by finding the greatest common factor (GCF), which was 2, and dividing both the numerator and the denominator by it. This gave us the simplified fraction 7/3.
Finally, we converted the improper fraction 7/3 into a mixed number to get a clearer sense of its value, resulting in 2 1/3.
So, the final answer to the question
