To add the fractions [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] and express the result as a mixed number in its simplest form, follow these steps:
### Step 1: Find a common denominator
To add the fractions, we first need to find a common denominator. The denominators are 7 and 5. The least common multiple (LCM) of 7 and 5 is 35.
### Step 2: Convert to equivalent fractions
Next, we'll convert each fraction to an equivalent fraction with the common denominator of 35.
- For [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[
\frac{5}{7} = \frac{5 \times 5}{7 \times 5} = \frac{25}{35}
\][/tex]
- For [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35}
\][/tex]
### Step 3: Add the fractions
Now that the fractions have the same denominator, we can add them:
[tex]\[
\frac{25}{35} + \frac{28}{35} = \frac{25 + 28}{35} = \frac{53}{35}
\][/tex]
### Step 4: Convert the improper fraction to a mixed number
The fraction [tex]\(\frac{53}{35}\)[/tex] is an improper fraction (the numerator is larger than the denominator). To convert it to a mixed number, we divide the numerator by the denominator.
[tex]\[
53 \div 35 = 1 \quad \text{remainder} \quad 18
\][/tex]
So, [tex]\(\frac{53}{35}\)[/tex] can be written as:
[tex]\[
1 \frac{18}{35}
\][/tex]
### Step 5: Simplify if possible
Lastly, we check if [tex]\(\frac{18}{35}\)[/tex] can be simplified further. Both 18 and 35 do not have any common factors other than 1, so the fraction is already in its simplest form.
### Final Answer
The sum of [tex]\(\frac{5}{7}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] is:
[tex]\[
1 \frac{18}{35}
\][/tex]
