To solve the fraction [tex]\(\frac{2 - \frac{1}{3}}{3 + \frac{1}{2}}\)[/tex]:
1. Simplify the numerator:
[tex]\[
2 - \frac{1}{3}
\][/tex]
First, convert 2 to a fraction with a common denominator:
[tex]\[
2 = \frac{6}{3}
\][/tex]
Now subtract:
[tex]\[
\frac{6}{3} - \frac{1}{3} = \frac{6 - 1}{3} = \frac{5}{3}
\][/tex]
So, the simplified numerator is:
[tex]\[
\frac{5}{3} = 1.6666666666666667
\][/tex]
2. Simplify the denominator:
[tex]\[
3 + \frac{1}{2}
\][/tex]
First, convert 3 to a fraction with a common denominator:
[tex]\[
3 = \frac{6}{2}
\][/tex]
Now add:
[tex]\[
\frac{6}{2} + \frac{1}{2} = \frac{6 + 1}{2} = \frac{7}{2}
\][/tex]
So, the simplified denominator is:
[tex]\[
\frac{7}{2} = 3.5
\][/tex]
3. Calculate the fraction:
[tex]\[
\frac{ \frac{5}{3}}{\frac{7}{2}}
\][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[
\frac{5}{3} \div \frac{7}{2} = \frac{5}{3} \times \frac{2}{7} = \frac{5 \times 2}{3 \times 7} = \frac{10}{21}
\][/tex]
So the value of the overall fraction is:
[tex]\[
\frac{10}{21} = 0.4761904761904762
\][/tex]
4. Compare this result with the given options:
[tex]\[
1. \ \frac{6}{35} \approx 0.17142857142857143
\][/tex]
[tex]\[
2. \ \frac{10}{21} \approx 0.47619047619047616
\][/tex]
[tex]\[
3. \ 5 \frac{5}{6} = 5 + \frac{5}{6} = 5 + 0.8333333333333333 = 5.833333333333333
\][/tex]
[tex]\[
4. \ 2 \frac{1}{10} = 2 + \frac{1}{10} = 2 + 0.1 = 2.1
\][/tex]
By comparing the numerical results, the correct answer is:
[tex]\[
\boxed{\frac{10}{21}}
\][/tex]
