Adding Fractions: -5/12 + 7/4 Solution

Hey guys! Let's break down this fraction problem step by step. We're going to tackle the addition of $-\frac{5}{12}$ and $\frac{7}{4}$. It might look a bit tricky at first, but don't worry, we'll get through it together!

Finding the Smallest Common Denominator

Okay, so before we can add these fractions, we need to find the smallest common denominator (LCD). What's a common denominator, you ask? It's a number that both denominators (the bottom numbers of the fractions) can divide into evenly. In our case, we have 12 and 4 as our denominators. So, what's the smallest number that both 12 and 4 can divide into without leaving a remainder?

Think about the multiples of 4: 4, 8, 12, 16, and so on. Notice that 12 is in this list! And, of course, 12 can divide into itself. So, the smallest common denominator here is 12. This is super important because it allows us to rewrite the fractions with the same denominator, making addition a breeze.

Why is finding the smallest one important? Well, we could use a larger common denominator (like 24, for example), but using the smallest one keeps the numbers smaller and easier to work with. It also means we'll have less simplifying to do at the end. Trust me, your future self will thank you for finding the smallest common denominator!

To recap, the smallest common denominator for the fractions $-\frac{5}{12}$ and $\frac{7}{4}$ is 12. Make sure you understand this concept because it is the foundation for easily adding and subtracting fractions.

Rewriting the Problem with Common Denominators

Alright, now that we've found our smallest common denominator (which is 12), let's rewrite our fractions so they both have this denominator. The first fraction, $-\frac{5}{12}$, already has 12 as its denominator, so we don't need to change it. It stays as $-\frac{5}{12}$.

Now, let's look at the second fraction, $\frac{7}{4}$. We need to turn the denominator 4 into 12. To do this, we think: "What do I need to multiply 4 by to get 12?" The answer is 3, because 4 multiplied by 3 equals 12.

But here's the golden rule: whatever you do to the bottom (the denominator), you must do to the top (the numerator). So, we multiply both the numerator and the denominator of $\frac{7}{4}$ by 3:

$\frac{7}{4} \times \frac{3}{3} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12}$

So, $\frac{7}{4}$ is equivalent to $\frac{21}{12}$. Now we can rewrite our original problem with the common denominator:

$-\frac{5}{12} + \frac{7}{4} = -\frac{5}{12} + \frac{21}{12}$

See how much easier that looks? When fractions have the same denominator, we can simply add (or subtract) the numerators. Having a common denominator allows us to accurately combine the fractions, kind of like comparing apples to apples instead of apples to oranges. Remember, this step is crucial for solving the problem correctly!

Solving the Problem

Okay, we've found the smallest common denominator (12), and we've rewritten our problem with the common denominators. Now it's time to actually solve the problem. We have:

$-\frac{5}{12} + \frac{21}{12}$

Since the denominators are the same, we can simply add the numerators. Remember that we're adding a negative number to a positive number. Think of it like this: you owe someone 5 dollars (-5) and you have 21 dollars (+21). After paying back what you owe, how much money do you have left?

So, we add the numerators: -5 + 21. This is the same as 21 - 5, which equals 16.

Therefore, $-\frac{5}{12} + \frac{21}{12} = \frac{16}{12}$

But wait, we're not quite done yet! It's always a good idea to simplify your answer if possible.

Simplifying the Answer

We've arrived at the answer $\frac{16}{12}$, but we can simplify this fraction. To simplify, we need to find the greatest common factor (GCF) of the numerator (16) and the denominator (12). The GCF is the largest number that divides evenly into both numbers.

The factors of 16 are: 1, 2, 4, 8, and 16. The factors of 12 are: 1, 2, 3, 4, 6, and 12.

The greatest common factor of 16 and 12 is 4.

Now we divide both the numerator and the denominator by 4:

$\frac{16}{12} \div \frac{4}{4} = \frac{16 \div 4}{12 \div 4} = \frac{4}{3}$

So, $\frac{16}{12}$ simplifies to $\frac{4}{3}$. This is an improper fraction (the numerator is larger than the denominator), and we can convert it to a mixed number.

To convert $\frac{4}{3}$ to a mixed number, we divide 4 by 3. 3 goes into 4 one time with a remainder of 1. So, the mixed number is 1 and $\frac{1}{3}$ (1$\frac{1}{3}$).

Therefore, the final answer is $\frac{4}{3}$ or 1$\frac{1}{3}$. Great job, guys! You successfully added the fractions and simplified the result!

Summary

Let's quickly recap what we did:

1. Found the smallest common denominator (12).
2. Rewrote the fractions with the common denominator: $-\frac{5}{12} + \frac{21}{12}$.
3. Added the numerators: -5 + 21 = 16, giving us $\frac{16}{12}$.
4. Simplified the fraction $\frac{16}{12}$ to $\frac{4}{3}$ or 1$\frac{1}{3}$.

Remember these steps, and you'll be a fraction-adding pro in no time! Keep practicing, and don't be afraid to ask for help when you need it. You got this!

Practice Problems

Want to test your skills? Here are a couple of practice problems for you to try:

1. $-\frac{3}{8} + \frac{5}{2} = ?$
2. $\frac{1}{6} + \frac{7}{9} = ?$

Work through these problems using the steps we covered, and check your answers. Good luck!