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Solve the following expression:

[tex]\[ \frac{2 - \frac{1}{3}}{3 + \frac{1}{2}} = \][/tex]

A. [tex]\(\frac{6}{35}\)[/tex]
B. [tex]\(\frac{10}{21}\)[/tex]
C. [tex]\(5 \frac{5}{6}\)[/tex]
D. [tex]\(2 \frac{1}{10}\)[/tex]


Sagot :

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]