Adding Mixed Numbers: Step-by-Step Solutions & Verification

by SLV Team 60 views

Hey guys! Today, we're diving into the world of mixed numbers and how to add them together. We'll break down each step and even check our answers to make sure everything is perfect. So, grab your pencils, and let's get started!

Understanding Mixed Numbers

Before we jump into adding, let's quickly recap what mixed numbers actually are. A mixed number is simply a whole number combined with a fraction. For example, 2 rac{1}{4} is a mixed number, where 2 is the whole number part and rac{1}{4} is the fractional part. Understanding this basic concept is crucial for successfully adding mixed numbers.

Why are mixed numbers important? Well, they're super useful in everyday life. Imagine you're baking a cake and the recipe calls for 2 rac{1}{2} cups of flour. You wouldn't say 2.5 cups, would you? Using mixed numbers helps us represent quantities more clearly and intuitively, especially when dealing with fractions of things.

Now, let's talk about the strategy for adding mixed numbers. There are generally two approaches you can take:

  1. Convert to Improper Fractions: This involves turning each mixed number into an improper fraction (where the numerator is greater than or equal to the denominator) and then adding the fractions. Finally, convert the result back to a mixed number.
  2. Add Whole Numbers and Fractions Separately: This method involves adding the whole number parts together and the fractional parts together. If the sum of the fractions is an improper fraction, you'll need to convert it to a mixed number and add the whole number part to the sum of the whole numbers. We'll be using this method in the examples below.

Regardless of the method you choose, the key is to ensure you have a common denominator before adding the fractions. This means that the bottom numbers of the fractions need to be the same. If they aren't, you'll need to find the least common multiple (LCM) of the denominators and adjust the fractions accordingly.

Let's tackle some examples to see how this works in practice!

Example a) 2 rac{1}{4}+2 rac{3}{8}=4 rac{5}{8}

In this first example, we're adding 2 rac{1}{4} and 2 rac{3}{8}. The provided solution shows a clear step-by-step process, which is fantastic for understanding the mechanics of adding mixed numbers.

First, the fractions need to have a common denominator. We can see that rac{1}{4} can be easily converted to a fraction with a denominator of 8. How do we do that? We multiply both the numerator (1) and the denominator (4) by 2. This gives us rac{2}{8}. So, 2 rac{1}{4} becomes 2 rac{2}{8}. Getting a common denominator is important because you cannot add or subtract fractions without one.

Now, we can rewrite the problem as 2 rac{2}{8} + 2 rac{3}{8}. Next, we add the whole numbers together: 2 + 2 = 4. Then, we add the fractions: rac{2}{8} + rac{3}{8} = rac{5}{8}. Combining these, we get 4 rac{5}{8}. The final answer to this example is 4 rac{5}{8}.

But how do we check if this answer is correct? There are a couple of ways. One way is to convert the mixed numbers to improper fractions and add them. Another way is to use a calculator that can handle fractions. Let's use the improper fraction method for verification:

  • 2 rac{1}{4} converted to an improper fraction is rac{(2 imes 4) + 1}{4} = rac{9}{4}
  • 2 rac{3}{8} converted to an improper fraction is rac{(2 imes 8) + 3}{8} = rac{19}{8}

Now, we need a common denominator to add these improper fractions. We can convert rac{9}{4} to rac{18}{8} by multiplying both the numerator and denominator by 2. So, our problem becomes rac{18}{8} + rac{19}{8} = rac{37}{8}.

Finally, we convert rac{37}{8} back to a mixed number. 37 divided by 8 is 4 with a remainder of 5. So, rac{37}{8} is equal to 4 rac{5}{8}. This confirms that our initial solution was correct! Always double-check your work guys, especially when dealing with fractions.

Example b) 4 rac{2}{9}+2 rac{4}{3}=4 rac{2}{9}+2 rac{12}{27}=

Okay, let's tackle the second example: 4 rac{2}{9} + 2 rac{4}{3}. This one is a bit trickier, but we'll break it down step-by-step. The first part of the given solution starts well, attempting to find a common denominator. However, there's a little hiccup in the process that we need to address.

The initial step correctly identifies that we need a common denominator to add the fractions rac{2}{9} and rac{4}{3}. The solution attempts to convert rac{4}{3} to a fraction with a denominator of 27, resulting in 2 rac{12}{27}. While it's not wrong, it is not efficient. We can use the LCM (Least Common Multiple) to easily find a common denominator.

The mistake here is not finding the least common denominator. While 27 is a common multiple of 9 and 3, it's not the smallest. The LCM of 9 and 3 is actually 9. This means we only need to convert rac{4}{3} to a fraction with a denominator of 9. To do this, we multiply both the numerator and denominator of rac{4}{3} by 3, giving us rac{12}{9}.

So, we rewrite the problem as 4 rac{2}{9} + 2 rac{12}{9}. Now, let's add the whole numbers: 4 + 2 = 6. Then, we add the fractions: rac{2}{9} + rac{12}{9} = rac{14}{9}. This gives us 6 rac{14}{9}.

But wait, we're not quite done yet! The fraction rac{14}{9} is an improper fraction, meaning the numerator is larger than the denominator. We need to convert this to a mixed number. 14 divided by 9 is 1 with a remainder of 5. So, rac{14}{9} is equal to 1 rac{5}{9}.

Now, we add the whole number part of this mixed number (1) to the whole number part of our previous result (6): 6 + 1 = 7. This leaves us with the final answer: 7 rac{5}{9}. Therefore, the final and correct answer to this example is 7 rac{5}{9}.

To verify, let’s convert everything to improper fractions:

  • 4 rac{2}{9} = rac{(4 imes 9) + 2}{9} = rac{38}{9}
  • 2 rac{4}{3} = rac{(2 imes 3) + 4}{3} = rac{10}{3}

To add these, we need a common denominator, which is 9. Convert rac{10}{3} to rac{30}{9} (multiply numerator and denominator by 3).

Now add: rac{38}{9} + rac{30}{9} = rac{68}{9}.

Convert back to a mixed number: 68 divided by 9 is 7 with a remainder of 5. So, rac{68}{9} = 7 rac{5}{9}. This confirms our solution!

This example highlights the importance of simplifying fractions and using the least common denominator to make calculations easier. It also reinforces the need to convert improper fractions to mixed numbers in the final answer. This is crucial to show your results in their simplest form.

Key Takeaways for Adding Mixed Numbers

Before we wrap up, let's quickly summarize the key things to remember when adding mixed numbers:

  1. Find a Common Denominator: This is the golden rule of fraction addition. Make sure all fractions have the same denominator before you add them.
  2. Add Whole Numbers and Fractions Separately: This simplifies the process and makes it less prone to errors.
  3. Convert Improper Fractions: If the sum of the fractions is an improper fraction, convert it to a mixed number.
  4. Simplify: Always simplify your final answer as much as possible. This means reducing fractions to their lowest terms.
  5. Verify Your Answer: Double-checking your work, whether by converting to improper fractions or using a calculator, is always a good idea.

By following these steps, you'll be adding mixed numbers like a pro in no time! Math can be challenging, but practicing the fundamentals helps a lot.

Conclusion

So, there you have it! We've covered the ins and outs of adding mixed numbers, from finding common denominators to verifying our answers. Remember, practice makes perfect, so keep working at it, and you'll master this skill in no time. If you guys have any questions, feel free to ask, and happy adding!