Adding Mixed Numbers: Solve $13 \frac{5}{7} + \frac{13}{28}$
Hey guys! Today, we're diving into the wonderful world of mixed number addition. Specifically, we're going to tackle the problem of adding $13 \frac{5}{7}$ and $ rac{13}{28}$. It might seem a bit daunting at first, but trust me, we'll break it down step-by-step, and you'll be adding mixed numbers like a pro in no time. So, grab your pencils and paper, and let's get started!
Understanding Mixed Numbers
Before we jump into the addition, let's quickly recap what mixed numbers are. A mixed number is simply a combination of a whole number and a fraction. In our problem, $13 \frac{5}{7}$ is a mixed number, where 13 is the whole number part and $ rac{5}{7}$ is the fractional part. Understanding this basic structure is key to successfully adding mixed numbers.
Think of it like having 13 whole pizzas and then an extra $ rac{5}{7}$ of another pizza. Makes you hungry, right? But in math terms, it's just a way to represent a number that's greater than a whole but not quite the next whole number. Now that we're all on the same page about what mixed numbers are, let's move on to the actual addition process.
Finding a Common Denominator
The first crucial step in adding fractions (and mixed numbers) is to ensure they have a common denominator. Remember, the denominator is the bottom number in a fraction, and it tells us how many equal parts the whole is divided into. To add fractions, these parts need to be the same size. In our case, we need a common denominator for $ rac{5}{7}$ and $ rac{13}{28}$.
So, how do we find this common denominator? The easiest way is to look for the least common multiple (LCM) of the denominators, which are 7 and 28. What's the LCM of 7 and 28, you ask? Well, the multiples of 7 are 7, 14, 21, 28, and so on. And hey, look at that! 28 is a multiple of both 7 and 28 (obviously!). Therefore, 28 is our common denominator. This means we need to convert $ rac{5}{7}$ into an equivalent fraction with a denominator of 28.
To do this, we need to figure out what to multiply 7 by to get 28. The answer is 4 (7 * 4 = 28). Now, we multiply both the numerator (top number) and the denominator of $ rac{5}{7}$ by 4. This gives us $ rac{5 * 4}{7 * 4} = \frac{20}{28}$. So, now we have $ rac{20}{28}$ instead of $ rac{5}{7}$, and both fractions have the same denominator! This is a major step towards solving our problem. With a common denominator in hand, we're ready to move on to the next phase of the addition.
Adding the Fractions and Whole Numbers
Now that we've got a common denominator, the fun part begins: adding the fractions! We've transformed our original problem, $13 \frac{5}{7} + \frac{13}{28}$, into $13 \frac{20}{28} + \frac{13}{28}$. See how much simpler it looks already?
To add mixed numbers, we handle the whole numbers and the fractions separately. First, let's focus on the fractions. We have $ rac20}{28} + \frac{13}{28}$. Since they have the same denominator, we can simply add the numerators{28}$.
Next, we deal with the whole numbers. In this case, we only have one whole number, which is 13. So, for now, we just keep it aside. Now we have 13 and $\frac{33}{28}$. But wait! $\frac{33}{28}$ is an improper fraction because the numerator (33) is larger than the denominator (28). This means it's greater than 1, and we need to simplify it further. So, let's convert this improper fraction into a mixed number.
Converting Improper Fractions to Mixed Numbers
We've arrived at an improper fraction, $ rac{33}{28}$, and to make our final answer look its best, we need to convert it into a mixed number. Remember, an improper fraction has a numerator that is greater than or equal to the denominator. To convert, we simply divide the numerator by the denominator.
So, we divide 33 by 28. 28 goes into 33 one time (1 * 28 = 28), with a remainder of 5 (33 - 28 = 5). This means that $ rac{33}{28}$ is equal to 1 whole and $ rac{5}{28}$ leftover. Therefore, $\frac{33}{28} = 1 \frac{5}{28}$.
Now, remember that 13 we kept aside earlier? It's time to bring it back into the picture. We now have 13 + $1 \frac5}{28}$. We simply add the whole numbers together{28}$. So, the final mixed number is $14 \frac{5}{28}$. We're almost there! There's just one more little step to consider.
Simplifying the Fraction (If Possible)
We've reached $14 \frac{5}{28}$, but before we declare victory, we need to check if the fraction part, $ rac{5}{28}$, can be simplified. Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common factor (GCF).
So, what's the GCF of 5 and 28? Well, 5 is a prime number, meaning its only factors are 1 and itself. 28 is not divisible by 5. This means the GCF of 5 and 28 is 1. Since we can only divide both the numerator and denominator by 1, the fraction is already in its simplest form! Therefore, $ rac{5}{28}$ cannot be simplified further.
The Final Answer
Phew! We've made it through all the steps, and now we can confidently state our final answer. The sum of $13 \frac{5}{7} + \frac{13}{28}$ is $14 \frac{5}{28}$. That’s it! Wasn't so bad once we broke it down, right? You've successfully added mixed numbers, found common denominators, converted improper fractions, and simplified your answer. Give yourself a pat on the back!
Practice Makes Perfect
Like anything in math, the key to mastering adding mixed numbers is practice. Try working through some more examples on your own, and don't be afraid to make mistakes. Mistakes are just learning opportunities in disguise!
Remember the steps we followed today: find a common denominator, add the fractions and whole numbers separately, convert improper fractions to mixed numbers, and simplify if possible. Keep these steps in mind, and you'll be tackling even the trickiest mixed number problems with ease. Keep up the great work, guys! You've got this!