To add the mixed numbers [tex]\( 1 \frac{2}{3} \)[/tex] and [tex]\( 5 \frac{3}{8} \)[/tex], we can follow these steps:
1. Convert the mixed numbers to improper fractions:
[tex]\[
1 \frac{2}{3} = \frac{5}{3} \quad \text{(1 whole and 2/3 to improper fraction: \( 1 \times 3 + 2 = 5 \) over 3)}
\][/tex]
[tex]\[
5 \frac{3}{8} = \frac{43}{8} \quad \text{(5 whole and 3/8 to improper fraction: \( 5 \times 8 + 3 = 43 \) over 8)}
\][/tex]
2. Find a common denominator for the two fractions. The least common multiple (LCM) of 3 and 8 is 24.
3. Convert the fractions to have the common denominator:
[tex]\[
\frac{5}{3} = \frac{5 \times 8}{3 \times 8} = \frac{40}{24}
\][/tex]
[tex]\[
\frac{43}{8} = \frac{43 \times 3}{8 \times 3} = \frac{129}{24}
\][/tex]
4. Add the fractions:
[tex]\[
\frac{40}{24} + \frac{129}{24} = \frac{169}{24}
\][/tex]
5. Convert the improper fraction back to a mixed number. Divide 169 by 24:
[tex]\[
169 \div 24 = 7 \quad \text{(whole number)}
\][/tex]
The remainder is:
[tex]\[
169 - 7 \times 24 = 169 - 168 = 1
\][/tex]
So the remainder is 1, making the fraction part [tex]\(\frac{1}{24}\)[/tex].
Therefore, the final result in mixed number format is:
[tex]\[
7 \frac{1}{24}
\][/tex]
So, the correct answer is [tex]\( 7 \frac{1}{24} \)[/tex].
