Sure, let's evaluate the expression step-by-step:
Given expression:
[tex]\[
\frac{3}{2} \div \left( \frac{1}{4} + \frac{4}{7} \right)
\][/tex]
First, we need to find the sum inside the parentheses:
[tex]\[
\frac{1}{4} + \frac{4}{7}
\][/tex]
To add these two fractions, we need a common denominator. The least common multiple of 4 and 7 is 28.
Convert each fraction to have the denominator of 28:
[tex]\[
\frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28}
\][/tex]
[tex]\[
\frac{4}{7} = \frac{4 \times 4}{7 \times 4} = \frac{16}{28}
\][/tex]
Now add the fractions:
[tex]\[
\frac{7}{28} + \frac{16}{28} = \frac{7 + 16}{28} = \frac{23}{28}
\][/tex]
So, the sum inside the parentheses is:
[tex]\[
\frac{1}{4} + \frac{4}{7} = \frac{23}{28}
\][/tex]
Next, we need to divide [tex]\(\frac{3}{2}\)[/tex] by [tex]\(\frac{23}{28}\)[/tex]:
[tex]\[
\frac{3}{2} \div \frac{23}{28}
\][/tex]
Division by a fraction is the same as multiplication by its reciprocal:
[tex]\[
\frac{3}{2} \div \frac{23}{28} = \frac{3}{2} \times \frac{28}{23}
\][/tex]
Multiply the fractions:
[tex]\[
\frac{3 \times 28}{2 \times 23} = \frac{84}{46}
\][/tex]
Simplify the fraction by finding the greatest common divisor of the numerator and the denominator, which is 2:
[tex]\[
\frac{84 \div 2}{46 \div 2} = \frac{42}{23}
\][/tex]
So the final answer is:
[tex]\[
\frac{3}{2} \div \left( \frac{1}{4} + \frac{4}{7} \right) = \frac{42}{23}
\][/tex]
