To solve the expression [tex]\(\frac{3}{2} + \frac{5}{7} - \frac{2}{5}\)[/tex], we need to follow the steps for performing addition and subtraction of fractions. Here is the detailed, step-by-step solution:
1. Identify the fractions involved:
[tex]\[
\frac{3}{2}, \quad \frac{5}{7}, \quad \text{and} \quad \frac{2}{5}
\][/tex]
2. Find the Least Common Denominator (LCD) for the fractions.
The denominators are [tex]\(2\)[/tex], [tex]\(7\)[/tex], and [tex]\(5\)[/tex]. The LCD of these numbers is the smallest number that all three denominators divide into without leaving a remainder. The LCD of [tex]\(2\)[/tex], [tex]\(7\)[/tex], and [tex]\(5\)[/tex] is [tex]\(70\)[/tex].
3. Convert each fraction to an equivalent fraction with the denominator equal to the LCD:
- Convert [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
\frac{3}{2} = \frac{3 \times 35}{2 \times 35} = \frac{105}{70}
\][/tex]
- Convert [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[
\frac{5}{7} = \frac{5 \times 10}{7 \times 10} = \frac{50}{70}
\][/tex]
- Convert [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
\frac{2}{5} = \frac{2 \times 14}{5 \times 14} = \frac{28}{70}
\][/tex]
4. Perform the addition and subtraction with the equivalent fractions:
[tex]\[
\frac{105}{70} + \frac{50}{70} - \frac{28}{70}
\][/tex]
5. Add and subtract the numerators while keeping the denominator the same:
[tex]\[
\frac{105 + 50 - 28}{70} = \frac{127}{70}
\][/tex]
Hence, the result of the expression [tex]\(\frac{3}{2} + \frac{5}{7} - \frac{2}{5}\)[/tex] is:
[tex]\[
\frac{127}{70}
\][/tex]
