Sure! Let's break this down step by step:
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers [tex]\(2 \frac{2}{3}\)[/tex] and [tex]\(3 \frac{1}{2}\)[/tex] to improper fractions.
For [tex]\(2 \frac{2}{3}\)[/tex]:
- The whole number part is [tex]\(2\)[/tex].
- The fractional part is [tex]\(\frac{2}{3}\)[/tex].
- Convert the mixed number to an improper fraction:
[tex]\[
2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{6+2}{3} = \frac{8}{3}
\][/tex]
For [tex]\(3 \frac{1}{2}\)[/tex]:
- The whole number part is [tex]\(3\)[/tex].
- The fractional part is [tex]\(\frac{1}{2}\)[/tex].
- Convert the mixed number to an improper fraction:
[tex]\[
3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{6+1}{2} = \frac{7}{2}
\][/tex]
### Step 2: Find a Common Denominator
To add these fractions, we need a common denominator. The denominators are 3 and 2. The least common multiple of 3 and 2 is 6.
Convert the fractions to have the common denominator of 6:
[tex]\[
\frac{8}{3} = \frac{8 \times 2}{3 \times 2} = \frac{16}{6}
\][/tex]
[tex]\[
\frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6}
\][/tex]
### Step 3: Add the Fractions
Now that the fractions have a common denominator, we can add them:
[tex]\[
\frac{16}{6} + \frac{21}{6} = \frac{16 + 21}{6} = \frac{37}{6}
\][/tex]
### Step 4: Convert the Improper Fraction Back to a Mixed Number
Convert [tex]\(\frac{37}{6}\)[/tex] back to a mixed number:
- Divide [tex]\(37\)[/tex] by [tex]\(6\)[/tex], which gives us [tex]\(6\)[/tex] whole parts because [tex]\(6 \times 6 = 36\)[/tex] and a remainder of [tex]\(1\)[/tex].
- So, [tex]\(\frac{37}{6} = 6 \frac{1}{6}\)[/tex].
### Final Answer
The sum of [tex]\(2 \frac{2}{3}\)[/tex] and [tex]\(3 \frac{1}{2}\)[/tex] is:
[tex]\[
6 \frac{1}{6}
\][/tex]
Hence, the final result of [tex]\(2 \frac{2}{3} + 3 \frac{1}{2}\)[/tex] is [tex]\(6 \frac{1}{6}\)[/tex].
