To solve the given problem, which is to find the sum of [tex]\(5 \frac{3}{8} + 2 \frac{1}{4}\)[/tex], we can follow these steps:
1. Convert the mixed numbers to improper fractions:
- For [tex]\(5 \frac{3}{8}\)[/tex]:
- [tex]\(5\)[/tex] is the whole number part, and [tex]\(\frac{3}{8}\)[/tex] is the fractional part.
- Convert [tex]\(5\)[/tex] to a fraction with denominator [tex]\(8\)[/tex]:
[tex]\(5 = \frac{5 \cdot 8}{8} = \frac{40}{8}\)[/tex]
- Now add the fractional part:
[tex]\[
5 \frac{3}{8} = \frac{40}{8} + \frac{3}{8} = \frac{43}{8}
\][/tex]
- For [tex]\(2 \frac{1}{4}\)[/tex]:
- [tex]\(2\)[/tex] is the whole number part, and [tex]\(\frac{1}{4}\)[/tex] is the fractional part.
- Convert [tex]\(2\)[/tex] to a fraction with denominator [tex]\(4\)[/tex]:
[tex]\(2 = \frac{2 \cdot 4}{4} = \frac{8}{4}\)[/tex]
- Now add the fractional part:
[tex]\[
2 \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}
\][/tex]
2. Find a common denominator to add the fractions:
- The fractions we have are [tex]\(\frac{43}{8}\)[/tex] and [tex]\(\frac{9}{4}\)[/tex].
- The least common multiple (LCM) of [tex]\(8\)[/tex] and [tex]\(4\)[/tex] is [tex]\(8\)[/tex].
3. Adjust the numerators to the common denominator:
- For [tex]\(\frac{43}{8}\)[/tex], the numerator remains [tex]\(43\)[/tex] because the denominator is already [tex]\(8\)[/tex]:
[tex]\[
\frac{43}{8} = \frac{43 \cdot 1}{8} = \frac{43}{8}
\][/tex]
- For [tex]\(\frac{9}{4}\)[/tex], multiply both the numerator and the denominator by [tex]\(2\)[/tex] to get the common denominator of [tex]\(8\)[/tex]:
[tex]\[
\frac{9}{4} = \frac{9 \cdot 2}{4 \cdot 2} = \frac{18}{8}
\][/tex]
4. Add the fractions:
- Now we add the adjusted numerators:
[tex]\[
\frac{43}{8} + \frac{18}{8} = \frac{43 + 18}{8} = \frac{61}{8}
\][/tex]
5. Simplify if possible:
- The fraction [tex]\(\frac{61}{8}\)[/tex] is already in its simplest form because [tex]\(61\)[/tex] and [tex]\(8\)[/tex] have no common divisors other than [tex]\(1\)[/tex].
Therefore, the sum of [tex]\(5 \frac{3}{8} + 2 \frac{1}{4}\)[/tex] is:
[tex]\[
\boxed{\frac{61}{8}}
\][/tex]
