To determine the sign of the quotient for the expression \(-1 \frac{3}{4} \div \frac{2}{5}\), we first need to rewrite \(-1 \frac{3}{4}\) as an improper fraction.
The mixed number \(-1 \frac{3}{4}\) can be converted to an improper fraction as follows:
- Multiply the whole number part by the denominator: \(1 \times 4 = 4\).
- Add the numerator to that result: \(4 + 3 = 7\).
- Since the whole number is negative, we make it negative: \(-7\).
Thus, \(-1 \frac{3}{4} = -\frac{7}{4}\).
Now, we need to divide \(-\frac{7}{4}\) by \(\frac{2}{5}\). Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the division as follows:
\[ -\frac{7}{4} \div \frac{2}{5} = -\frac{7}{4} \times \frac{5}{2} \]
Now we can multiply the fractions:
\[ -\frac{7 \times 5}{4 \times 2} = -\frac{35}{8} \]
The quotient is \(-\frac{35}{8}\), which is a negative number.
Thus, the sign of the quotient is negative.