Sure! Let's solve the given expression step-by-step:
[tex]\[
\frac{2}{3} + \frac{1}{4} \times \frac{2}{5} - \frac{3}{4} + \frac{5}{2}
\][/tex]
Let's break this down into individual steps.
1. Evaluate [tex]\(\frac{2}{3}\)[/tex]:
This is simply a fraction:
[tex]\[
\frac{2}{3}
\][/tex]
2. Evaluate [tex]\(\frac{1}{4} \times \frac{2}{5}\)[/tex]:
Multiply the numerators and denominators:
[tex]\[
\frac{1 \times 2}{4 \times 5} = \frac{2}{20} = \frac{1}{10}
\][/tex]
3. Evaluate [tex]\(-\frac{3}{4}\)[/tex]:
This remains as is:
[tex]\[
-\frac{3}{4}
\][/tex]
4. Evaluate [tex]\(\frac{5}{2}\)[/tex]:
This is also a simple fraction:
[tex]\[
\frac{5}{2}
\][/tex]
Now let's combine these values step-by-step.
First, let’s add [tex]\(\frac{2}{3}\)[/tex] and the result from [tex]\(\frac{1}{4} \times \frac{2}{5}\)[/tex]:
[tex]\[
\frac{2}{3} + \frac{1}{10}
\][/tex]
To add these fractions, find a common denominator. The least common multiple of 3 and 10 is 30:
[tex]\[
\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}
\][/tex]
[tex]\[
\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}
\][/tex]
[tex]\[
\frac{20}{30} + \frac{3}{30} = \frac{23}{30}
\][/tex]
Next, subtract [tex]\(\frac{3}{4}\)[/tex] from the result:
[tex]\[
\frac{23}{30} - \frac{3}{4}
\][/tex]
The common denominator for 30 and 4 is 60:
[tex]\[
\frac{23}{30} = \frac{23 \times 2}{30 \times 2} = \frac{46}{60}
\][/tex]
[tex]\[
\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}
\][/tex]
[tex]\[
\frac{46}{60} - \frac{45}{60} = \frac{1}{60}
\][/tex]
Finally, let’s add [tex]\(\frac{5}{2}\)[/tex]:
[tex]\[
\frac{1}{60} + \frac{5}{2}
\][/tex]
The common denominator for 60 and 2 is 60:
[tex]\[
\frac{5}{2} = \frac{5 \times 30}{2 \times 30} = \frac{150}{60}
\][/tex]
[tex]\[
\frac{1}{60} + \frac{150}{60} = \frac{151}{60}
\][/tex]
Putting this result into decimal to verify it matches the value from the solution:
[tex]\[
\frac{151}{60} \approx 2.5166666666666666
\][/tex]
So, the answer is not one of the listed choices (A-D), thus there might be an error in the provided options or statement interpretation. Make sure that question is accurate. Based on the provided breakdown, answer is around [tex]\(\frac{151}{60}\)[/tex].
