To add the mixed numbers [tex]\(2 \frac{1}{4}\)[/tex] and [tex]\(2 \frac{2}{3}\)[/tex], we follow these detailed steps:
1. Convert each mixed number to an improper fraction:
- For [tex]\(2 \frac{1}{4}\)[/tex]:
[tex]\[
2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}
\][/tex]
- For [tex]\(2 \frac{2}{3}\)[/tex]:
[tex]\[
2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}
\][/tex]
2. Determine the common denominator:
- The denominators are 4 and 3. The least common multiple (LCM) of 4 and 3 is 12.
3. Convert each fraction to have the common denominator of 12:
- For [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[
\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}
\][/tex]
- For [tex]\(\frac{8}{3}\)[/tex]:
[tex]\[
\frac{8}{3} = \frac{8 \times 4}{3 \times 4} = \frac{32}{12}
\][/tex]
4. Add the fractions:
- [tex]\[
\frac{27}{12} + \frac{32}{12} = \frac{27 + 32}{12} = \frac{59}{12}
\][/tex]
5. Convert the improper fraction back to a mixed number:
- [tex]\[
\frac{59}{12} = 4 \frac{11}{12}
\][/tex]
- Here [tex]\(59\)[/tex] divided by [tex]\(12\)[/tex] gives a quotient of [tex]\(4\)[/tex], and the remainder is [tex]\(11\)[/tex]. So, [tex]\( \frac{59}{12} = 4 \frac{11}{12} \)[/tex].
Therefore, the sum of [tex]\(2 \frac{1}{4}\)[/tex] and [tex]\(2 \frac{2}{3}\)[/tex] is [tex]\(4 \frac{11}{12}\)[/tex].
