To solve the problem [tex]\(-\frac{5}{6} \div \left(-\frac{1}{3}\right)\)[/tex], we must follow the steps for dividing fractions. Specifically, dividing one fraction by another is equivalent to multiplying the first fraction by the reciprocal of the second fraction. Let’s go through each step in detail:
1. Identify the fractions involved:
- The first fraction is [tex]\(-\frac{5}{6}\)[/tex].
- The second fraction is [tex]\(-\frac{1}{3}\)[/tex].
2. Find the reciprocal of the second fraction:
- The reciprocal of [tex]\(-\frac{1}{3}\)[/tex] is [tex]\(-3\)[/tex]. In fraction form, the reciprocal is [tex]\(-\frac{3}{1}\)[/tex].
3. Convert the division problem into a multiplication problem:
[tex]\[-\frac{5}{6} \div \left(-\frac{1}{3}\right) = -\frac{5}{6} \times \left(-\frac{3}{1}\right)\][/tex]
4. Multiply the numerators and the denominators:
- Numerator multiplication: [tex]\((-5) \times (-3) = 15\)[/tex]
- Denominator multiplication: [tex]\(6 \times 1 = 6\)[/tex]
Therefore,
[tex]\[-\frac{5}{6} \times \left(-\frac{3}{1}\right) = \frac{15}{6}\][/tex]
5. Simplify the resulting fraction:
- Divide both the numerator and the denominator by their greatest common divisor. Since 15 and 6 have a common factor of 3, we divide both by 3:
[tex]\[\frac{15 \div 3}{6 \div 3} = \frac{5}{2}\][/tex]
6. Check the sign:
- Since both original fractions were negative, their product is positive, and thus the final fraction remains positive.
After simplification, the final answer is [tex]\(\boxed{\frac{5}{2}}\)[/tex].
