To subtract the mixed numbers [tex]\(8 \frac{1}{6} - 4 \frac{5}{6}\)[/tex], follow these steps:
1. Convert each mixed number to an improper fraction.
- For [tex]\( 8 \frac{1}{6} \)[/tex]:
- The whole number part is 8.
- The fractional part is [tex]\(\frac{1}{6}\)[/tex].
- To convert it to an improper fraction, multiply the whole number by the denominator and add the numerator:
[tex]\[
8 \frac{1}{6} = 8 + \frac{1}{6} = \frac{8 \times 6 + 1}{6} = \frac{48 + 1}{6} = \frac{49}{6}.
\][/tex]
- For [tex]\( 4 \frac{5}{6} \)[/tex]:
- The whole number part is 4.
- The fractional part is [tex]\(\frac{5}{6}\)[/tex].
- To convert it to an improper fraction, multiply the whole number by the denominator and add the numerator:
[tex]\[
4 \frac{5}{6} = 4 + \frac{5}{6} = \frac{4 \times 6 + 5}{6} = \frac{24 + 5}{6} = \frac{29}{6}.
\][/tex]
2. Subtract the improper fractions.
- Both fractions have the same denominator, so subtract the numerators directly:
[tex]\[
\frac{49}{6} - \frac{29}{6} = \frac{49 - 29}{6} = \frac{20}{6}.
\][/tex]
3. Simplify the resulting fraction.
- Simplify [tex]\(\frac{20}{6}\)[/tex] by finding the greatest common divisor (GCD) of 20 and 6, which is 2:
[tex]\[
\frac{20}{6} = \frac{20 \div 2}{6 \div 2} = \frac{10}{3}.
\][/tex]
4. Convert the simplified improper fraction back to a mixed number.
- Divide the numerator by the denominator:
[tex]\[
10 \div 3 = 3 \text{ with a remainder of } 1.
\][/tex]
- The quotient is the whole number part, and the remainder is the numerator of the fractional part:
[tex]\[
\frac{10}{3} = 3 \frac{1}{3}.
\][/tex]
Therefore, the result of subtracting [tex]\(8 \frac{1}{6} - 4 \frac{5}{6}\)[/tex] simplifies to [tex]\(3 \frac{1}{3}\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{3 \frac{1}{3}}. \][/tex]
