To simplify the given expression [tex]\(4 \frac{1}{2} \div 1 \frac{7}{8}\)[/tex], follow these steps:
1. Convert the mixed numbers to improper fractions:
- [tex]\(4 \frac{1}{2}\)[/tex] can be written as [tex]\(4 + \frac{1}{2}\)[/tex], which equals [tex]\(\frac{8}{2} + \frac{1}{2} = \frac{9}{2}\)[/tex].
- [tex]\(1 \frac{7}{8}\)[/tex] can be written as [tex]\(1 + \frac{7}{8}\)[/tex], which equals [tex]\(\frac{8}{8} + \frac{7}{8} = \frac{15}{8}\)[/tex].
2. Rewrite the division as a multiplication by the reciprocal of the divisor:
- [tex]\(\frac{9}{2} \div \frac{15}{8}\)[/tex] is equivalent to [tex]\(\frac{9}{2} \times \frac{8}{15}\)[/tex].
3. Multiply the fractions:
[tex]\[
\frac{9}{2} \times \frac{8}{15} = \frac{9 \times 8}{2 \times 15} = \frac{72}{30}.
\][/tex]
4. Simplify the resulting fraction by finding the greatest common divisor (GCD) of 72 and 30:
- The GCD of 72 and 30 is 6.
- Divide the numerator and the denominator by their GCD to reduce the fraction:
[tex]\[
\frac{72 \div 6}{30 \div 6} = \frac{12}{5}.
\][/tex]
5. Convert the improper fraction back to a mixed number:
- [tex]\(\frac{12}{5}\)[/tex] is equal to [tex]\(2 \frac{2}{5}\)[/tex] (since 12 divided by 5 is 2 with a remainder of 2, thus the remainder over the denominator forms the fractional part).
Thus, the simplified answer is [tex]\(2 \frac{2}{5}\)[/tex], which corresponds to option D.
