To divide \( 14 \frac{1}{2} \) by \( 1 \frac{7}{8} \), we first need to convert both mixed numbers into improper fractions.
- Convert \( 14 \frac{1}{2} \) to an improper fraction:
\[ 14 \frac{1}{2} = \frac{(14 \times 2) + 1}{2} = \frac{28 + 1}{2} = \frac{29}{2} \]
- Convert \( 1 \frac{7}{8} \) to an improper fraction:
\[ 1 \frac{7}{8} = \frac{(1 \times 8) + 7}{8} = \frac{8 + 7}{8} = \frac{15}{8} \]
- Now, we divide \( \frac{29}{2} \) by \( \frac{15}{8} \). Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{29}{2} \div \frac{15}{8} = \frac{29}{2} \times \frac{8}{15} \]
- Multiply the fractions:
\[ \frac{29 \times 8}{2 \times 15} = \frac{232}{30} \]
- Simplify the fraction:
To simplify \( \frac{232}{30} \), we can find the greatest common divisor (GCD) of 232 and 30. The GCD is 2, so we divide both the numerator and the denominator by 2:
\[ \frac{232 \div 2}{30 \div 2} = \frac{116}{15} \]
- Convert \( \frac{116}{15} \) back to a mixed number:
To convert \( \frac{116}{15} \):
- Divide 116 by 15, which goes 7 times with a remainder.
\[ 116 \div 15 = 7 \quad \text{(because } 15 \times 7 = 105\text{)} \]
Remainder:
\[ 116 - 105 = 11 \]
So, \( \frac{116}{15} = 7 \frac{11}{15} \).
Thus, the final result of \( 14 \frac{1}{2} \div 1 \frac{7}{8} \) is:
\[ \boxed{7 \frac{11}{15}} \]