Adding Fractions: Step-by-Step Solutions

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Hey guys! Let's dive into the world of fractions and tackle some addition problems. We'll break down each step, making it super easy to follow along. So, grab your pencils, and let's get started!

a) 2/3 + 5/4

When you're adding fractions, the first thing you absolutely must do is ensure they have a common denominator. Think of it like this: you can't add apples and oranges directly, right? You need a common unit, like "fruits." Similarly, fractions need a common denominator, which is the bottom number.

So, how do we find this common denominator? We look for the least common multiple (LCM) of the denominators we have. In this case, our denominators are 3 and 4. The multiples of 3 are 3, 6, 9, 12, 15, and so on. The multiples of 4 are 4, 8, 12, 16, and so on. Spot the common one? Yep, it's 12! So, 12 will be our common denominator.

Now, we need to convert each fraction to have this denominator. For 2/3, we ask ourselves: "What do we multiply 3 by to get 12?" The answer is 4. So, we multiply both the numerator (the top number) and the denominator by 4: (2 * 4) / (3 * 4) = 8/12.

Next, we do the same for 5/4. "What do we multiply 4 by to get 12?" The answer is 3. So, we multiply both the numerator and denominator by 3: (5 * 3) / (4 * 3) = 15/12.

Great! Now we have 8/12 + 15/12. Since the denominators are the same, we can simply add the numerators: 8 + 15 = 23. So, we have 23/12.

This is an improper fraction because the numerator is larger than the denominator. We can convert it to a mixed number. How many times does 12 go into 23? Once, with a remainder of 11. So, 23/12 is equal to 1 11/12. Boom! We solved our first fraction addition problem!

b) 11/1 + 3/8

Okay, let's tackle the next one: 11/1 + 3/8. Don't let the 11/1 freak you out! Remember that any whole number can be written as a fraction with a denominator of 1. So, 11/1 is just the same as 11.

Now, we need that common denominator. We have 1 and 8 as our denominators. What's the LCM of 1 and 8? Well, any number is a multiple of 1, so the LCM here is simply 8.

Let's convert our fractions. For 11/1, we need to multiply both the numerator and the denominator by 8 to get a denominator of 8: (11 * 8) / (1 * 8) = 88/8.

The second fraction, 3/8, already has the correct denominator, so we don't need to change it.

Now we can add: 88/8 + 3/8. Add the numerators: 88 + 3 = 91. So, we have 91/8.

Again, this is an improper fraction. Let's convert it to a mixed number. How many times does 8 go into 91? It goes 11 times (11 * 8 = 88), with a remainder of 3. So, 91/8 is equal to 11 3/8. Awesome!

c) 1/6 + 19/4

Moving on to c) 1/6 + 19/4. Let's find that common denominator again! Our denominators are 6 and 4. What's the LCM of 6 and 4? The multiples of 6 are 6, 12, 18, and so on. The multiples of 4 are 4, 8, 12, 16, and so on. We meet again, 12! That's our LCM.

Converting 1/6, we ask: "What do we multiply 6 by to get 12?" The answer is 2. Multiply both numerator and denominator by 2: (1 * 2) / (6 * 2) = 2/12.

For 19/4, we ask: "What do we multiply 4 by to get 12?" The answer is 3. Multiply both numerator and denominator by 3: (19 * 3) / (4 * 3) = 57/12.

Now we add: 2/12 + 57/12. Add the numerators: 2 + 57 = 59. So, we have 59/12.

Improper fraction alert! Let's convert to a mixed number. How many times does 12 go into 59? It goes 4 times (4 * 12 = 48), with a remainder of 11. So, 59/12 is equal to 4 11/12. You're getting the hang of this!

d) 5/12 + 7/24

Let's keep the fraction party going with d) 5/12 + 7/24. Time to find that common denominator! We have 12 and 24. What's the LCM of 12 and 24? Well, 24 is a multiple of 12 (12 * 2 = 24), so the LCM is simply 24.

Converting 5/12, we ask: "What do we multiply 12 by to get 24?" The answer is 2. Multiply both numerator and denominator by 2: (5 * 2) / (12 * 2) = 10/24.

The second fraction, 7/24, already has the correct denominator. Score!

Now we add: 10/24 + 7/24. Add the numerators: 10 + 7 = 17. So, we have 17/24.

This fraction is actually a proper fraction because the numerator (17) is smaller than the denominator (24). This means we can't simplify it to a mixed number. So, our answer is simply 17/24. Easy peasy!

e) 5/16 + 3/48

Last but not least, we have e) 5/16 + 3/48. Let's find that common denominator one more time! Our denominators are 16 and 48. What's the LCM of 16 and 48? Notice that 48 is a multiple of 16 (16 * 3 = 48), so the LCM is 48.

Converting 5/16, we ask: "What do we multiply 16 by to get 48?" The answer is 3. Multiply both numerator and denominator by 3: (5 * 3) / (16 * 3) = 15/48.

The second fraction, 3/48, is already good to go!

Now we add: 15/48 + 3/48. Add those numerators: 15 + 3 = 18. So, we have 18/48.

This is a proper fraction, but we can simplify it! Both 18 and 48 are divisible by 6. Divide both the numerator and denominator by 6: (18 / 6) / (48 / 6) = 3/8. And there we have it! Our final answer is 3/8.

Conclusion

And that's how you add fractions, guys! Remember the key is to find that common denominator, convert your fractions, add the numerators, and simplify if needed. You've got this! Keep practicing, and you'll be a fraction master in no time. 🚀