(2 3/8 - 1 5/6 ) : (- 1 5/8)

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:

\[ \boxed{ - \frac{1}{3} } \]