We have a complex fraction to evaluate. The numerator is seven halves minus one fourth, and the denominator is two thirds minus one fourth. Let's work through this step by step to find the answer.
First, let's calculate the numerator: seven halves minus one fourth. To subtract fractions, we need a common denominator. The least common denominator of 2 and 4 is 4. Converting seven halves to fourths gives us fourteen fourths. So fourteen fourths minus one fourth equals thirteen fourths.
Now let's calculate the denominator: two thirds minus one fourth. The least common denominator of 3 and 4 is 12. Converting two thirds to twelfths gives us eight twelfths, and one fourth becomes three twelfths. So eight twelfths minus three twelfths equals five twelfths.
Now we divide the numerator by the denominator: thirteen fourths divided by five twelfths. To divide fractions, we multiply by the reciprocal. So thirteen fourths times twelve fifths equals thirteen times twelve over four times five, which is one hundred fifty-six over twenty.
Finally, we simplify one hundred fifty-six over twenty. We can factor this as thirteen times twelve over four times five. Since twelve equals three times four, we have thirteen times three times four over four times five. We can cancel the common factor of four, leaving us with thirteen times three over five, which equals thirty-nine fifths. Therefore, the answer is E: thirty-nine fifths.