Fraction Sums: Step-by-Step Solutions

Hey guys! Let's break down some fraction addition problems. We'll tackle each one step by step, so it's super easy to follow. Adding fractions might seem tricky at first, but with a little practice, you'll be a pro in no time! We'll cover everything from simple fractions with common denominators to mixed numbers and absolute values. So grab your pencil and paper, and let's dive in!

a. ${ \frac{13}{5} + \frac{21}{5} }$

Let's start with the first expression: ${ \frac{13}{5} + \frac{21}{5} }$. When you're adding fractions, the first thing you want to check is whether they have the same denominator. In this case, both fractions have a denominator of 5, which makes our job way easier!

When the denominators are the same, you can simply add the numerators (the numbers on top) and keep the denominator the same. So, we have:

${ \frac{13}{5} + \frac{21}{5} = \frac{13 + 21}{5} = \frac{34}{5} }$

Now, let's simplify the fraction. Since 34 is larger than 5, we can convert this improper fraction (where the numerator is greater than the denominator) into a mixed number. To do this, we divide 34 by 5.

34 divided by 5 is 6 with a remainder of 4. This means that ${ \frac{34}{5} }$ is equal to 6 whole units and ${ \frac{4}{5} }$ left over. Therefore:

${ \frac{34}{5} = 6 \frac{4}{5} }$

So, the final answer for the first expression is ${ 6 \frac{4}{5} }$. Wasn't that easy? When the denominators are the same, it's just a matter of adding the numerators and simplifying!

b. ${ \frac{5}{6} + \frac{3}{8} }$

Okay, let's move on to the next one: ${ \frac{5}{6} + \frac{3}{8} }$. This time, the fractions have different denominators. We've got a 6 and an 8. To add these fractions, we need to find a common denominator. The easiest way to find a common denominator is to find the least common multiple (LCM) of the two denominators. The LCM is the smallest number that both denominators divide into evenly.

Let's list the multiples of 6 and 8 to find their LCM:

• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 8: 8, 16, 24, 32, ...

The smallest number that appears in both lists is 24. So, the LCM of 6 and 8 is 24. This means we need to convert both fractions so they have a denominator of 24.

To convert ${ \frac{5}{6} }$ to a fraction with a denominator of 24, we need to multiply both the numerator and the denominator by the same number. Since 6 times 4 is 24, we multiply both the numerator and denominator of ${ \frac{5}{6} }$ by 4:

${ \frac{5}{6} \times \frac{4}{4} = \frac{20}{24} }$

Similarly, to convert ${ \frac{3}{8} }$ to a fraction with a denominator of 24, we need to multiply both the numerator and the denominator by the same number. Since 8 times 3 is 24, we multiply both the numerator and denominator of ${ \frac{3}{8} }$ by 3:

${ \frac{3}{8} \times \frac{3}{3} = \frac{9}{24} }$

Now we can add the fractions, because they have the same denominator:

${ \frac{20}{24} + \frac{9}{24} = \frac{20 + 9}{24} = \frac{29}{24} }$

Finally, we can convert this improper fraction to a mixed number. 29 divided by 24 is 1 with a remainder of 5. So:

${ \frac{29}{24} = 1 \frac{5}{24} }$

The final answer is ${ 1 \frac{5}{24} }$. Remember, finding a common denominator is key when adding fractions with different denominators!

c. ${ \frac{3}{5} + 2 \frac{3}{5} }$

Next up, we've got ${ \frac{3}{5} + 2 \frac{3}{5} }$. This one involves a mixed number. There are a couple of ways we can approach this. One way is to convert the mixed number to an improper fraction first. Let's do that!

To convert ${ 2 \frac{3}{5} }$ to an improper fraction, we multiply the whole number (2) by the denominator (5) and add the numerator (3). This gives us the new numerator, and we keep the same denominator:

${ 2 \frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5} }$

Now we can rewrite the original expression as:

${ \frac{3}{5} + \frac{13}{5} }$

Since the denominators are the same, we can simply add the numerators:

${ \frac{3}{5} + \frac{13}{5} = \frac{3 + 13}{5} = \frac{16}{5} }$

Finally, we convert this improper fraction to a mixed number. 16 divided by 5 is 3 with a remainder of 1. So:

${ \frac{16}{5} = 3 \frac{1}{5} }$

Therefore, the final answer is ${ 3 \frac{1}{5} }$.

Alternatively, we could have added the fractional parts and the whole number parts separately. In this case, ${ \frac{3}{5} + \frac{3}{5} = \frac{6}{5} = 1 \frac{1}{5} }$. Then add the whole number 2 to get ${ 2+1 \frac{1}{5} = 3 \frac{1}{5} }$. This can be a faster approach if the numbers are simple enough!

d. ${ 2 \frac{1}{3} + \frac{3}{8} + 3 \frac{5}{6} }$

Alright, let's tackle this bigger one: ${ 2 \frac{1}{3} + \frac{3}{8} + 3 \frac{5}{6} }$. We've got mixed numbers and fractions with different denominators. First, let's convert the mixed numbers to improper fractions:

${ 2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} }$

${ 3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6} }$

Now we rewrite the expression as:

${ \frac{7}{3} + \frac{3}{8} + \frac{23}{6} }$

Now, we need to find a common denominator for 3, 8, and 6. Let's find the LCM of these numbers:

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
• Multiples of 8: 8, 16, 24, 32, ...
• Multiples of 6: 6, 12, 18, 24, 30, ...

The LCM of 3, 8, and 6 is 24. So we need to convert each fraction to have a denominator of 24:

${ \frac{7}{3} \times \frac{8}{8} = \frac{56}{24} }$

${ \frac{3}{8} \times \frac{3}{3} = \frac{9}{24} }$

${ \frac{23}{6} \times \frac{4}{4} = \frac{92}{24} }$

Now we can add the fractions:

${ \frac{56}{24} + \frac{9}{24} + \frac{92}{24} = \frac{56 + 9 + 92}{24} = \frac{157}{24} }$

Finally, convert this improper fraction to a mixed number. 157 divided by 24 is 6 with a remainder of 13. So:

${ \frac{157}{24} = 6 \frac{13}{24} }$

Therefore, the final answer is ${ 6 \frac{13}{24} }$.

e. ${ \left|-\frac{2}{5} + \frac{3}{8}\right| + \left|\frac{4}{7} + \frac{2}{7}\right| }$

Last but not least, we have ${ \left|-\frac{2}{5} + \frac{3}{8}\right| + \left|\frac{4}{7} + \frac{2}{7}\right| }$. This one involves absolute values, which might look intimidating, but don't worry, it's not too bad! Remember that the absolute value of a number is its distance from zero, so it's always non-negative.

First, let's evaluate the expressions inside the absolute value signs. For the first one, we have ${ -\frac{2}{5} + \frac{3}{8} }$. We need a common denominator for 5 and 8, which is 40.

${ -\frac{2}{5} \times \frac{8}{8} = -\frac{16}{40} }$

${ \frac{3}{8} \times \frac{5}{5} = \frac{15}{40} }$

So:

${ -\frac{16}{40} + \frac{15}{40} = -\frac{1}{40} }$

Now, for the second absolute value, we have ${ \frac{4}{7} + \frac{2}{7} }$. The denominators are already the same, so:

${ \frac{4}{7} + \frac{2}{7} = \frac{6}{7} }$

Now, we can rewrite the original expression as:

${ \left|-\frac{1}{40}\right| + \left|\frac{6}{7}\right| }$

The absolute value of ${ -\frac{1}{40} }$ is ${ \frac{1}{40} }$, and the absolute value of ${ \frac{6}{7} }$ is ${ \frac{6}{7} }$. So:

${ \frac{1}{40} + \frac{6}{7} }$

We need a common denominator for 40 and 7, which is 280.

${ \frac{1}{40} \times \frac{7}{7} = \frac{7}{280} }$

${ \frac{6}{7} \times \frac{40}{40} = \frac{240}{280} }$

So:

${ \frac{7}{280} + \frac{240}{280} = \frac{247}{280} }$

Therefore, the final answer is ${ \frac{247}{280} }$.

And there you have it! We've tackled a bunch of fraction addition problems, from simple ones with common denominators to more complex ones with mixed numbers and absolute values. Remember to always look for a common denominator when adding fractions, and don't be afraid of mixed numbers – just convert them to improper fractions. Keep practicing, and you'll become a fraction master in no time! You got this!