Adding Fractions Simplify $2/3 + 5/6$ Solution

Hey guys! Let's dive into solving this fraction addition problem and make sure we get the answer in its simplest form. We're tackling the problem: $2/3 + 5/6$. This is a classic math question that you might see in elementary or middle school, and it's super important to understand how to add fractions correctly. So, let's break it down step by step.

Understanding the Problem

When we look at fraction addition, the most important thing to remember is that we can only add fractions directly if they have the same denominator. The denominator is the bottom number in a fraction – it tells us how many parts the whole is divided into. In our case, we have $2/3$ and $5/6$. The denominators are 3 and 6, which are different. So, we can't just add the numerators (the top numbers) yet. We need to find a common denominator first. A common denominator is a number that both denominators can divide into evenly. For 3 and 6, the least common multiple (LCM) is 6. This means we want to convert both fractions so they have a denominator of 6. This is a crucial first step because it allows us to compare and combine the fractions accurately. Think of it like trying to add apples and oranges – you can't do it directly until you have a common unit, like fruit. By finding a common denominator, we're essentially giving our fractions a common unit so we can add them together. This foundation is key to mastering more complex fraction problems later on, so let's make sure we nail it!

Finding a Common Denominator

Okay, so we know we need a common denominator, and we've identified that 6 is the least common multiple of 3 and 6. That means we need to convert $2/3$ into an equivalent fraction with a denominator of 6. How do we do that? Well, we need to figure out what we can multiply the original denominator (3) by to get our new denominator (6). In this case, 3 multiplied by 2 equals 6. But here's the golden rule of fractions: Whatever you do to the bottom, you gotta do to the top! So, if we multiply the denominator (3) by 2, we also need to multiply the numerator (2) by 2. Let's do the math: $2 * 2 = 4$. So, $2/3$ becomes $4/6$. Now we have two fractions with the same denominator: $4/6$ and $5/6$. This step is absolutely crucial because it ensures we are adding equal parts of a whole. If we didn't have a common denominator, we'd be trying to add fractions that represent different-sized pieces, which would give us a wrong answer. By converting $2/3$ to $4/6$, we're making sure both fractions are speaking the same language, so to speak. It's like converting inches to feet before you add them together – you need a common unit to get an accurate result. Remember, finding the least common denominator often makes the problem easier to work with, especially when simplifying later on. So, mastering this step is a huge win!

Adding the Fractions

Now that we've got our fractions with a common denominator, the fun part begins – adding them! We have $4/6 + 5/6$. When you add fractions with the same denominator, you simply add the numerators and keep the denominator the same. Think of it like adding slices of a pie. If you have 4 slices out of 6, and you add 5 more slices out of 6, you'll have a total of 9 slices out of 6. So, let's add the numerators: $4 + 5 = 9$. And we keep the denominator the same, which is 6. So, $4/6 + 5/6 = 9/6$. We've got our answer, but we're not done yet! Remember, the question asks us to put the answer in its simplest form. This means we need to make sure the fraction is reduced as much as possible. $9/6$ is what we call an improper fraction because the numerator (9) is larger than the denominator (6). This means the fraction represents a value greater than 1. To simplify it, we can either convert it to a mixed number or reduce the fraction directly. We'll tackle that in the next step. But for now, give yourselves a pat on the back – you've successfully added the fractions! This is a big step in mastering fraction operations, and you're doing great. Keep in mind that this process is the foundation for many other math concepts, so understanding it well will pay off in the long run.

Simplifying the Answer

Alright, we've got $9/6$, which is our answer so far. But as we discussed, we need to simplify it. There are a couple of ways we can do this. First, let's look at reducing the fraction directly. We need to find the greatest common factor (GCF) of the numerator (9) and the denominator (6). The greatest common factor is the largest number that divides evenly into both numbers. The factors of 9 are 1, 3, and 9. The factors of 6 are 1, 2, 3, and 6. The largest number they have in common is 3. So, the GCF of 9 and 6 is 3. To reduce the fraction, we divide both the numerator and the denominator by the GCF. So, $9 ÷ 3 = 3$ and $6 ÷ 3 = 2$. This means $9/6$ reduces to $3/2$. Now, $3/2$ is still an improper fraction, so we can convert it to a mixed number. A mixed number has a whole number part and a fractional part. To convert $3/2$ to a mixed number, we divide the numerator (3) by the denominator (2). 3 divided by 2 is 1 with a remainder of 1. The quotient (1) becomes the whole number part of the mixed number, the remainder (1) becomes the new numerator, and the denominator (2) stays the same. So, $3/2$ is equal to 1 rac{1}{2}. This is the simplified form of our answer! We've taken an improper fraction and turned it into a mixed number, making it easier to understand and visualize. Great job on getting to the final answer!

The Final Answer

So, after all our hard work, we've found that 2/3 + 5/6 = 1 rac{1}{2}. This is the simplest form of our answer, and it matches option B from our choices. We started by finding a common denominator, then added the fractions, and finally simplified the result. This process is the key to adding fractions successfully every time. Remember, it's not just about getting the right answer; it's about understanding the steps and why they work. This knowledge will help you tackle more complex math problems in the future. You've learned how to add fractions with different denominators, reduce fractions to their simplest form, and convert between improper fractions and mixed numbers. That's a lot of math power! Keep practicing, and you'll become a fraction master in no time. Math can be challenging, but with a little bit of patience and a step-by-step approach, you can solve any problem. Give yourselves a big round of applause for sticking with it and reaching the solution! Now you can confidently say you know how to add fractions and simplify the answers. Keep up the fantastic work!

Therefore, the correct answer is:

B) 1 rac{1}{2}