To solve the addition of the mixed numbers [tex]\(3 \frac{5}{6}\)[/tex] and [tex]\(2 \frac{2}{3}\)[/tex], follow these steps:
1. Convert the mixed numbers to improper fractions:
- For [tex]\(3 \frac{5}{6}\)[/tex]:
[tex]\[
3 \frac{5}{6} = \frac{3 \cdot 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}
\][/tex]
- For [tex]\(2 \frac{2}{3}\)[/tex]:
[tex]\[
2 \frac{2}{3} = \frac{2 \cdot 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}
\][/tex]
2. Find a common denominator:
- The denominators of the fractions are 6 and 3. The least common multiple (LCM) of 6 and 3 is 6.
3. Convert fractions to have the same denominator:
- To convert [tex]\(\frac{8}{3}\)[/tex] to a denominator of 6:
[tex]\[
\frac{8}{3} = \frac{8 \cdot 2}{3 \cdot 2} = \frac{16}{6}
\][/tex]
4. Add the numerators:
- Now, add the fractions [tex]\(\frac{23}{6}\)[/tex] and [tex]\(\frac{16}{6}\)[/tex]:
[tex]\[
\frac{23}{6} + \frac{16}{6} = \frac{23 + 16}{6} = \frac{39}{6}
\][/tex]
5. Simplify the resulting fraction:
- The greatest common divisor (GCD) of 39 and 6 is 3. Simplify [tex]\(\frac{39}{6}\)[/tex]:
[tex]\[
\frac{39}{6} = \frac{39 \div 3}{6 \div 3} = \frac{13}{2}
\][/tex]
6. Convert back to a mixed number:
- To convert [tex]\(\frac{13}{2}\)[/tex] to a mixed number, divide the numerator by the denominator:
[tex]\[
13 \div 2 = 6 \text{ remainder } 1
\][/tex]
Thus, [tex]\(\frac{13}{2} = 6 \frac{1}{2}\)[/tex].
So, the simplified form of the sum [tex]\(3 \frac{5}{6} + 2 \frac{2}{3}\)[/tex] is:
[tex]\[
6 \frac{1}{2}
\][/tex]
