To solve the given expression step-by-step, we first need to address the multiplication within the expression, and then proceed with the addition and subtraction. Here's the expression we must solve:
[tex]\[
\frac{2}{3}+\frac{1}{4} \times \frac{2}{5}-\frac{3}{4}+\frac{5}{2}
\][/tex]
1. Calculate the multiplication:
[tex]\[
\frac{1}{4} \times \frac{2}{5} = \frac{1 \cdot 2}{4 \cdot 5} = \frac{2}{20} = \frac{1}{10}
\][/tex]
2. Substitute the multiplication result back into the expression:
[tex]\[
\frac{2}{3} + \frac{1}{10} - \frac{3}{4} + \frac{5}{2}
\][/tex]
3. Convert all fractions to a common denominator to simplify addition and subtraction:
- The least common multiple (LCM) of the denominators (3, 10, 4, and 2) is 60.
- Convert each fraction:
[tex]\[
\frac{2}{3} = \frac{2 \cdot 20}{3 \cdot 20} = \frac{40}{60}
\][/tex]
[tex]\[
\frac{1}{10} = \frac{1 \cdot 6}{10 \cdot 6} = \frac{6}{60}
\][/tex]
[tex]\[
\frac{3}{4} = \frac{3 \cdot 15}{4 \cdot 15} = \frac{45}{60}
\][/tex]
[tex]\[
\frac{5}{2} = \frac{5 \cdot 30}{2 \cdot 30} = \frac{150}{60}
\][/tex]
4. Now, perform the arithmetic operations with the common denominator:
[tex]\[
\frac{40}{60} + \frac{6}{60} - \frac{45}{60} + \frac{150}{60}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{40 + 6 - 45 + 150}{60} = \frac{151}{60}
\][/tex]
5. Convert the fraction result to a decimal (optional):
[tex]\[
\frac{151}{60} \approx 2.5167
\][/tex]
Thus, the simplified fraction result is:
[tex]\[
\boxed{\frac{151}{60}}
\][/tex]
Since none of the provided answer choices match [tex]\(\frac{151}{60}\)[/tex] or the approximate decimal value [tex]\(2.5167\)[/tex], it seems that none of the provided options are correct. The simplified exact answer is:
[tex]\[
\frac{151}{60}
\][/tex]
