Adding Fractions: What Is 2/3 + 2/9?

Hey guys! Let's dive into a super common math problem: adding fractions. Specifically, we're going to tackle the question: What is the sum of 2/3 and 2/9? This might seem tricky at first, but I promise it's totally manageable once you understand the basic steps. So, grab your pencils and let's get started!

Understanding the Basics of Fraction Addition

Before we jump into this specific problem, let's quickly recap the fundamentals of adding fractions. You can only directly add fractions if they have the same denominator, which is the bottom number of a fraction. The denominator tells you how many equal parts the whole is divided into. For instance, in the fraction 2/3, the denominator 3 means the whole is divided into three equal parts, and we have two of those parts. Similarly, in the fraction 2/9, the denominator 9 means the whole is divided into nine equal parts, and we have two of those parts. So, remember, equal denominators are key!

If the denominators are different, we need to find a common denominator before we can add the numerators (the top numbers). A common denominator is a number that both denominators can divide into evenly. The easiest way to find a common denominator is often to find the least common multiple (LCM) of the two denominators. The least common multiple is the smallest number that is a multiple of both denominators. Once you have a common denominator, you convert each fraction to an equivalent fraction with that denominator. An equivalent fraction represents the same value as the original fraction but has a different denominator. To convert a fraction, you multiply both the numerator and the denominator by the same number. This ensures that the value of the fraction remains unchanged.

Adding fractions with the same denominator is straightforward. You simply add the numerators and keep the denominator the same. For example, if you have 1/5 + 2/5, you add the numerators (1 + 2 = 3) and keep the denominator (5), resulting in 3/5. This is because you're adding like parts – in this case, fifths. After adding the fractions, always check if you can simplify the result. Simplifying a fraction means reducing it to its lowest terms. You do this by dividing both the numerator and the denominator by their greatest common factor (GCF). The greatest common factor is the largest number that divides evenly into both the numerator and the denominator. Simplifying fractions makes them easier to understand and work with. For instance, the fraction 4/8 can be simplified to 1/2 by dividing both the numerator and the denominator by 4.

Solving 2/3 + 2/9 Step-by-Step

Okay, now that we've refreshed our understanding of fraction addition, let's tackle the problem at hand: 2/3 + 2/9. The first thing we need to do is identify the denominators of the two fractions. In this case, the denominators are 3 and 9. Since the denominators are different, we need to find a common denominator before we can add the fractions. To find a common denominator, we can look for the least common multiple (LCM) of 3 and 9. The multiples of 3 are 3, 6, 9, 12, 15, and so on. The multiples of 9 are 9, 18, 27, 36, and so on. The smallest number that appears in both lists is 9, so the least common multiple of 3 and 9 is 9. This means that 9 is the common denominator we'll use to add the fractions. Now we need to convert each fraction to an equivalent fraction with a denominator of 9. The second fraction, 2/9, already has a denominator of 9, so we don't need to change it. However, we need to convert the first fraction, 2/3, to an equivalent fraction with a denominator of 9. To do this, we need to determine what number we need to multiply the denominator 3 by to get 9. Since 3 multiplied by 3 equals 9, we need to multiply both the numerator and the denominator of 2/3 by 3. So, 2/3 becomes (2 * 3) / (3 * 3) = 6/9.

Now that we have both fractions with the same denominator, we can add them. We have 6/9 + 2/9. To add these fractions, we add the numerators and keep the denominator the same. So, 6 + 2 = 8, and the denominator remains 9. Therefore, 6/9 + 2/9 = 8/9. The final step is to check if the resulting fraction can be simplified. In this case, the fraction 8/9 cannot be simplified because 8 and 9 do not have any common factors other than 1. This means that 8/9 is already in its simplest form. Therefore, the sum of 2/3 and 2/9 is 8/9. And that's it! You've successfully added the fractions. Remember the key steps: find a common denominator, convert the fractions, add the numerators, and simplify if possible. With a little practice, you'll become a fraction-adding pro in no time!

Why Finding a Common Denominator Matters

You might be wondering, why all the fuss about finding a common denominator? Why can't we just add the numerators and denominators separately? Well, the reason is that fractions represent parts of a whole, and we can only add parts that are of the same size. Think of it like trying to add apples and oranges – they're different things, so you can't simply say you have a certain number of