To solve the problem of adding the fractions [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] and simplifying the resulting fraction, follow these steps:
### Step 1: Find a common denominator
When adding fractions, we need to have a common denominator. The least common multiple (LCM) of the denominators 3 and 5 is 15. Therefore, we will convert both fractions to equivalent fractions with a common denominator of 15.
### Step 2: Convert the fractions
We convert each fraction to have the common denominator of 15:
- For [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}
\][/tex]
- For [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}
\][/tex]
### Step 3: Add the fractions
Now that both fractions have the same denominator, we can add the numerators while keeping the common denominator:
[tex]\[
\frac{10}{15} + \frac{12}{15} = \frac{10 + 12}{15} = \frac{22}{15}
\][/tex]
### Step 4: Simplify the fraction
The fraction [tex]\(\frac{22}{15}\)[/tex] is already in its simplest form because the greatest common divisor (GCD) of 22 and 15 is 1.
### Conclusion
The sum of the fractions [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{4}{5}\)[/tex] is:
[tex]\[
\frac{2}{3} + \frac{4}{5} = \frac{22}{15}
\][/tex]
Therefore, the final simplified fraction is [tex]\(\frac{22}{15}\)[/tex].
Note that [tex]\(\frac{10}{12}\)[/tex] was mentioned in the question, but the correct sum of the given fractions after adding and simplifying is indeed [tex]\(\frac{22}{15}\)[/tex].
