To determine if any of the given sums result in a rational number, let's evaluate each one carefully. A rational number can be expressed as the quotient of two integers, a/b, where a and b are integers and b is not zero.
Let's examine each sum:
### Option (A): [tex]\(\frac{1}{2} + 2\pi\)[/tex]
- The first term is [tex]\(\frac{1}{2}\)[/tex], which is rational.
- The second term is [tex]\(2\pi\)[/tex]. Since [tex]\(\pi\)[/tex] is an irrational number, multiplying it by 2 still results in an irrational number.
- Adding a rational number ([tex]\(\frac{1}{2}\)[/tex]) to an irrational number ([tex]\(2\pi\)[/tex]) always gives an irrational number.
Thus, [tex]\(\frac{1}{2} + 2\pi\)[/tex] is irrational.
### Option (B): [tex]\(\frac{\pi}{2} + 2\pi\)[/tex]
- The first term is [tex]\(\frac{\pi}{2}\)[/tex], which is irrational.
- The second term is [tex]\(2\pi\)[/tex]. Since [tex]\(\pi\)[/tex] is irrational, [tex]\(2\pi\)[/tex] is also irrational.
- Adding two irrational numbers, in general, results in an irrational number. Specifically, in this case, simplifying the expression gives [tex]\(\frac{\pi + 4\pi}{2} = \frac{5\pi}{2}\)[/tex], which is also irrational.
Thus, [tex]\(\frac{\pi}{2} + 2\pi\)[/tex] is irrational.
### Option (C): [tex]\(\frac{2\pi}{5\pi} + \frac{1}{2}\)[/tex]
- The first term simplifies to [tex]\(\frac{2\pi}{5\pi}\)[/tex], which simplifies to [tex]\(\frac{2}{5}\)[/tex], a rational number.
- The second term is [tex]\(\frac{1}{2}\)[/tex], which is also rational.
- Adding two rational numbers results in another rational number, [tex]\(\frac{2}{5} + \frac{1}{2} = \frac{4}{10} + \frac{5}{10} = \frac{9}{10}\)[/tex], which is rational.
So, [tex]\(\frac{2\pi}{5\pi} + \frac{1}{2}\)[/tex] simplifies to a rational number, [tex]\(\frac{9}{10}\)[/tex].
### Option (D): [tex]\(\frac{2}{\pi} + \frac{\pi}{2}\)[/tex]
- The first term is [tex]\(\frac{2}{\pi}\)[/tex], which is irrational (since dividing a rational number by an irrational number results in an irrational number).
- The second term is [tex]\(\frac{\pi}{2}\)[/tex], which is irrational.
- Adding two irrational numbers generally results in an irrational number.
Thus, [tex]\(\frac{2}{\pi} + \frac{\pi}{2}\)[/tex] is irrational.
Given this analysis, the only expression that results in a rational number is:
[tex]\[ \boxed{\mathrm{(C)} \ \frac{2\pi}{5\pi}+\frac{1}{2}} \][/tex]
