To determine which quotient is between -4 and -5 on a number line, let's calculate each expression:
-
\(2 \frac{5}{6} \div \left(-\frac{1}{2}\right)\):
- Convert \(2 \frac{5}{6}\) to an improper fraction: \[ 2 \frac{5}{6} = \frac{12}{6} + \frac{5}{6} = \frac{17}{6} \]
- Now divide: \[ \frac{17}{6} \div \left(-\frac{1}{2}\right) = \frac{17}{6} \times \left(-2\right) = \frac{17 \times -2}{6} = \frac{-34}{6} = -\frac{17}{3} \approx -5.67 \]
-
\(-5 \frac{1}{2} \div \left(-1 \frac{1}{2}\right)\):
- Convert \(-5 \frac{1}{2}\) to an improper fraction: \[ -5 \frac{1}{2} = -\frac{11}{2} \]
- Convert \(-1 \frac{1}{2}\) to an improper fraction: \[ -1 \frac{1}{2} = -\frac{3}{2} \]
- Now divide: \[ -\frac{11}{2} \div -\frac{3}{2} = -\frac{11}{2} \times -\frac{2}{3} = \frac{11 \times 2}{2 \times 3} = \frac{11}{3} \approx 3.67 \]
-
\(-9 \frac{1}{3} \div 2\):
- Convert \(-9 \frac{1}{3}\) to an improper fraction: \[ -9 \frac{1}{3} = -\frac{28}{3} \]
- Now divide: \[ -\frac{28}{3} \div 2 = -\frac{28}{3} \times \frac{1}{2} = -\frac{28}{6} = -\frac{14}{3} \approx -4.67 \]
-
\(1 \frac{2}{3} \div \left(-\frac{1}{2}\right)\):
- Convert \(1 \frac{2}{3}\) to an improper fraction: \[ 1 \frac{2}{3} = \frac{5}{3} \]
- Now divide: \[ \frac{5}{3} \div \left(-\frac{1}{2}\right) = \frac{5}{3} \times \left(-2\right) = \frac{5 \times -2}{3} = -\frac{10}{3} \approx -3.33 \]
Now, let's summarize our calculations and check which one is between -4 and -5:
- \(-\frac{17}{3} \approx -5.67\) (this quotient is less than -5)
- \(\frac{11}{3} \approx 3.67\) (this quotient is positive)
- \(-\frac{14}{3} \approx -4.67\) (this quotient is between -4 and -5)
- \(-\frac{10}{3} \approx -3.33\) (this quotient is greater than -4)
The only quotient that lies between -4 and -5 is: \[ -\frac{14}{3} \quad \text{(which is } -9 \frac{1}{3} \div 2\text{)} \]
Final answer is: \( -9 \frac{1}{3} \div 2\)