To simplify the given expression:
[tex]\[
\frac{2 + \frac{1}{6}}{2 - \frac{5}{6}}
\][/tex]
follow these steps:
1. Simplify the numerator:
[tex]\[
2 + \frac{1}{6}
\][/tex]
Convert the integer 2 to a fraction with a denominator of 6:
[tex]\[
2 = \frac{12}{6}
\][/tex]
Now, add the fractions:
[tex]\[
2 + \frac{1}{6} = \frac{12}{6} + \frac{1}{6} = \frac{13}{6}
\][/tex]
So the numerator is:
[tex]\[
2 + \frac{1}{6} = \frac{13}{6}
\][/tex]
2. Simplify the denominator:
[tex]\[
2 - \frac{5}{6}
\][/tex]
Convert the integer 2 to a fraction with a denominator of 6:
[tex]\[
2 = \frac{12}{6}
\][/tex]
Now, subtract the fractions:
[tex]\[
2 - \frac{5}{6} = \frac{12}{6} - \frac{5}{6} = \frac{7}{6}
\][/tex]
So the denominator is:
[tex]\[
2 - \frac{5}{6} = \frac{7}{6}
\][/tex]
3. Combine the fractions:
Now, you have the simplified numerator and denominator:
[tex]\[
\frac{\frac{13}{6}}{\frac{7}{6}}
\][/tex]
To simplify this complex fraction, multiply the numerator by the reciprocal of the denominator:
[tex]\[
\frac{13}{6} \times \frac{6}{7}
\][/tex]
The 6s cancel out:
[tex]\[
\frac{13}{7}
\][/tex]
So, the simplified result is:
[tex]\[
\frac{13}{7}
\][/tex]
4. Convert to decimal form:
To express this as a decimal:
[tex]\[
\frac{13}{7} \approx 1.857142857142857
\][/tex]
Therefore, the fully simplified fraction in its decimal form is approximately:
[tex]\[
1.857142857142857
\][/tex]
Thus, the simplified form of the fraction [tex]\(\frac{2 + \frac{1}{6}}{2 - \frac{5}{6}}\)[/tex] is [tex]\( \frac{13}{7} \)[/tex], which is approximately equal to [tex]\( 1.857142857142857 \)[/tex] in decimal form.
