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 follow the steps in order, carefully performing multiplication, addition, and subtraction of fractions.
1. Perform the multiplication [tex]\(\frac{1}{4} \times \frac{2}{5}\)[/tex]:
[tex]\[\frac{1 \times 2}{4 \times 5} = \frac{2}{20} = \frac{1}{10}\][/tex]
2. Rewrite the original expression with the result of the multiplication:
[tex]\[\frac{2}{3} + \frac{1}{10} - \frac{3}{4} + \frac{5}{2}\][/tex]
3. Find a common denominator for all fractions to perform addition and subtraction:
The denominators are 3, 10, 4, and 2. The least common multiple (LCM) of these denominators will be used:
- LCM of 3 and 10: 30
- LCM of 30 and 4: 60
- LCM of 60 and 2: 60
Thus, the common denominator is 60.
4. Convert all fractions to have the common denominator 60:
[tex]\[
\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}
\][/tex]
[tex]\[
\frac{1}{10} = \frac{1 \times 6}{10 \times 6} = \frac{6}{60}
\][/tex]
[tex]\[
\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}
\][/tex]
[tex]\[
\frac{5}{2} = \frac{5 \times 30}{2 \times 30} = \frac{150}{60}
\][/tex]
5. Now we perform the addition and subtraction:
[tex]\[
\frac{2}{3} + \frac{1}{10} - \frac{3}{4} + \frac{5}{2} = \frac{40}{60} + \frac{6}{60} - \frac{45}{60} + \frac{150}{60}
\][/tex]
[tex]\[
= \frac{40 + 6 - 45 + 150}{60}
\][/tex]
[tex]\[
= \frac{151}{60}
\][/tex]
6. Simplify the result, if possible:
The greatest common divisor (GCD) of 151 and 60 is 1, so the fraction cannot be simplified further.
Therefore:
[tex]\[
\frac{2}{3} + \frac{1}{4} \times \frac{2}{5} - \frac{3}{4} + \frac{5}{2} = \frac{151}{60}
\][/tex]
Since [tex]\(\frac{151}{60}\)[/tex] does not need further simplification and does not match any simplified fractions given in the answer choices (since it is also an improper fraction), [tex]\(\frac{151}{60}\)[/tex] is the correct and final form.
However, none of the provided multiple-choice answers accurately reflect this result. Therefore, the correct answer is [tex]\(\frac{151}{60}\)[/tex].
