Let's simplify the expression step by step:
[
\left( 2 \frac{3}{8} - 1 \frac{5}{6} \right) : \left( - 1 \frac{5}{8} \right)
]
Step 1: Convert mixed numbers to improper fractions
( 2 \frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8} )
( 1 \frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{6 + 5}{6} = \frac{11}{6} )
( - 1 \frac{5}{8} = - \frac{1 \times 8 + 5}{8} = - \frac{8 + 5}{8} = - \frac{13}{8} )
Step 2: Subtract the fractions
Find a common denominator for (\frac{19}{8}) and (\frac{11}{6}).
- The least common denominator (LCD) of 8 and 6 is 24.
Convert:
[
\frac{19}{8} = \frac{19 \times 3}{8 \times 3} = \frac{57}{24}
]
[
\frac{11}{6} = \frac{11 \times 4}{6 \times 4} = \frac{44}{24}
]
Subtract:
[
\frac{57}{24} - \frac{44}{24} = \frac{13}{24}
]
Step 3: Divide by (- \frac{13}{8})
Division of fractions is multiplication by the reciprocal:
[
\frac{13}{24} \div \left( - \frac{13}{8} \right) = \frac{13}{24} \times \left( - \frac{8}{13} \right)
]
Simplify:
- The 13 in numerator and denominator cancels out:
[
= 1 \times \left( - \frac{8}{24} \right) = - \frac{8}{24}
]
Reduce:
[
- \frac{8}{24} = - \frac{1}{3}
]
Final answer:
[
\boxed{ - \frac{1}{3} }
]
