Adding Fractions: Step-by-Step Solutions

Hey guys! Let's dive into the world of fractions and tackle some addition problems. Adding fractions might seem a bit tricky at first, but once you get the hang of it, it's a piece of cake! We'll go through each problem step by step, so you can follow along and understand exactly how to solve them. Ready? Let's get started!

a) $\frac{2}{3} + \frac{5}{4}$

When it comes to adding fractions, the golden rule is that you need a common denominator. This means the bottom numbers of the fractions (the denominators) have to be the same. In this case, we have $\frac{2}{3}$ and $\frac{5}{4}$. The denominators are 3 and 4. So, what's the smallest number that both 3 and 4 can divide into evenly? That's right, it's 12! This is our least common denominator (LCD).

Now, we need to convert both fractions to have this denominator. For $\frac{2}{3}$, 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:

$\frac{2}{3} * \frac{4}{4} = \frac{8}{12}$

Next, we do the same for $\frac{5}{4}$. "What do we multiply 4 by to get 12?" The answer is 3. So, we multiply both the numerator and the denominator by 3:

$\frac{5}{4} * \frac{3}{3} = \frac{15}{12}$

Now that both fractions have the same denominator, we can add them:

$\frac{8}{12} + \frac{15}{12} = \frac{8 + 15}{12} = \frac{23}{12}$

So, $\frac{2}{3} + \frac{5}{4} = \frac{23}{12}$. This is an improper fraction (the numerator is larger than the denominator), and we can leave it like that, or we can convert it to a mixed number. To do that, we divide 23 by 12. 12 goes into 23 once, with a remainder of 11. So, the mixed number is $1\frac{11}{12}$. Voilà! We solved the first one!

b) $\frac{11}{1} + \frac{3}{8}$

Okay, let's move on to the next one! This time we have $\frac{11}{1} + \frac{3}{8}$. Don't let the $\frac{11}{1}$ throw you off – it's just another way of writing the whole number 11. Think of it as 11 whole pizzas, each cut into 1 slice (silly, I know, but it works!).

So, we need to find a common denominator for 1 and 8. What's the smallest number that both 1 and 8 divide into evenly? You guessed it – it's 8. This makes things pretty simple because $\frac{3}{8}$ already has the denominator we need. We just need to convert $\frac{11}{1}$ to have a denominator of 8.

To do this, we multiply both the numerator and the denominator of $\frac{11}{1}$ by 8:

$\frac{11}{1} * \frac{8}{8} = \frac{88}{8}$

Now we can add the fractions:

$\frac{88}{8} + \frac{3}{8} = \frac{88 + 3}{8} = \frac{91}{8}$

So, $\frac{11}{1} + \frac{3}{8} = \frac{91}{8}$. Again, this is an improper fraction. Let's convert it to a mixed number. How many times does 8 go into 91? Well, 8 goes into 91 eleven times (8 * 11 = 88), with a remainder of 3. So, the mixed number is $11\frac{3}{8}$. Easy peasy!

c) $\frac{1}{6} + \frac{19}{9}$

Alright, let's tackle this one: $\frac{1}{6} + \frac{19}{9}$. Here, our denominators are 6 and 9. Finding the least common denominator (LCD) for 6 and 9 requires a bit of thought. Let's list some multiples of each:

• Multiples of 6: 6, 12, 18, 24, 30...
• Multiples of 9: 9, 18, 27, 36...

The smallest number that appears in both lists is 18. So, 18 is our LCD.

Now, let's convert our fractions. For $\frac{1}{6}$, we need to multiply both the numerator and the denominator by 3 (because 6 * 3 = 18):

$\frac{1}{6} * \frac{3}{3} = \frac{3}{18}$

For $\frac{19}{9}$, we need to multiply both the numerator and the denominator by 2 (because 9 * 2 = 18):

$\frac{19}{9} * \frac{2}{2} = \frac{38}{18}$

Now, we can add them:

$\frac{3}{18} + \frac{38}{18} = \frac{3 + 38}{18} = \frac{41}{18}$

So, $\frac{1}{6} + \frac{19}{9} = \frac{41}{18}$. Once again, we have an improper fraction. Let's convert it to a mixed number. 18 goes into 41 twice (18 * 2 = 36), with a remainder of 5. So, the mixed number is $2\frac{5}{18}$. Nailed it! We're on a roll here, guys!

d) $\frac{5}{12} + \frac{7}{24}$

Let's keep the ball rolling with $\frac{5}{12} + \frac{7}{24}$. This one is a little easier because 24 is a multiple of 12. That means our least common denominator is simply 24. We only need to convert $\frac{5}{12}$ to have a denominator of 24.

To do this, we multiply both the numerator and the denominator of $\frac{5}{12}$ by 2 (because 12 * 2 = 24):

$\frac{5}{12} * \frac{2}{2} = \frac{10}{24}$

Now we can add the fractions:

$\frac{10}{24} + \frac{7}{24} = \frac{10 + 7}{24} = \frac{17}{24}$

So, $\frac{5}{12} + \frac{7}{24} = \frac{17}{24}$. This time, the fraction is already in its simplest form, and it's a proper fraction (the numerator is smaller than the denominator), so we don't need to convert it to a mixed number. Sweet!

e) $\frac{5}{16} + \frac{3}{48}$

Last but not least, let's tackle $\frac{5}{16} + \frac{3}{48}$. Similar to the previous problem, 48 is a multiple of 16 (16 * 3 = 48), so our least common denominator is 48. We only need to convert $\frac{5}{16}$ to have a denominator of 48.

To do this, we multiply both the numerator and the denominator of $\frac{5}{16}$ by 3:

$\frac{5}{16} * \frac{3}{3} = \frac{15}{48}$

Now we can add the fractions:

$\frac{15}{48} + \frac{3}{48} = \frac{15 + 3}{48} = \frac{18}{48}$

So, $\frac{5}{16} + \frac{3}{48} = \frac{18}{48}$. But wait! This fraction can be simplified. Both 18 and 48 are divisible by 6. So, let's divide both the numerator and the denominator by 6:

$\frac{18 ÷ 6}{48 ÷ 6} = \frac{3}{8}$

Therefore, $\frac{5}{16} + \frac{3}{48} = \frac{3}{8}$. Fantastic! We simplified it to its simplest form.

Conclusion

And there you have it, folks! We've successfully added all the fractions, step by step. Remember, the key to adding fractions is to find a common denominator, convert the fractions, add the numerators, and simplify if possible. With a little practice, you'll be a fraction-adding pro in no time!

I hope this was helpful and easy to understand. Keep practicing, and you'll master these skills in no time. Until next time, happy calculating! You got this! Remember practice makes perfect. Do not be afraid of fractions. Adding them is actually fun!