To solve the expression [tex]\( \frac{2}{3} + \frac{1}{4} \times \frac{2}{5} - \frac{3}{4} + \frac{5}{2} \)[/tex], we need to proceed step-by-step, simplifying the fractions and performing each arithmetic operation in sequence.
1. First Term:
[tex]\[
\frac{2}{3} \approx 0.6667
\][/tex]
2. Second Term (Multiplication):
[tex]\[
\frac{1}{4} \times \frac{2}{5} = \frac{1 \times 2}{4 \times 5} = \frac{2}{20} = \frac{1}{10} = 0.1
\][/tex]
3. Third Term:
[tex]\[
\frac{3}{4} = 0.75
\][/tex]
4. Fourth Term:
[tex]\[
\frac{5}{2} = 2.5
\][/tex]
Next, we will perform the addition and subtraction in the correct order:
5. Addition of the First and Second Terms:
[tex]\[
\frac{2}{3} + \frac{1}{10} \approx 0.6667 + 0.1 = 0.7667
\][/tex]
6. Subtraction of the Third Term:
[tex]\[
0.7667 - \frac{3}{4} \approx 0.7667 - 0.75 = 0.0167
\][/tex]
7. Addition of the Fourth Term:
[tex]\[
0.0167 + \frac{5}{2} \approx 0.0167 + 2.5 = 2.5167
\][/tex]
Therefore, the final value of the expression [tex]\( \frac{2}{3} + \frac{1}{4} \times \frac{2}{5} - \frac{3}{4} + \frac{5}{2} \)[/tex] is approximately [tex]\( 2.5167 \)[/tex].
Given the options, none of these directly match the calculated result exactly. However, this numerical approach shows that the closest match to our result within the options is not there. If this is a multiple-choice question with the given options:
- A. [tex]\(-\frac{67}{50}\left(\frac{68}{51} \times \frac{65}{60} \frac{50}{52}+\right.\)[/tex]
- B. [tex]\(\frac{13}{15}\)[/tex]
- C. [tex]\(-\frac{23}{50}\)[/tex]
None of them match the calculated result [tex]\(2.5167\)[/tex]. Therefore, an error might be present in either the question narrative or the options provided.
