The sum of twice a fraction and its reciprocal is 17/6. If the fraction, in lowest terms, has numerator 3, determine the fraction by forming and solving the appropriate equation.

Introduction / Context:
This aptitude question involves fractions, reciprocals, and forming a linear equation from a word statement. The condition links twice a fraction and its reciprocal to a given rational number. An extra piece of information tells us that the fraction, in lowest terms, has numerator 3. You must translate the statement into algebra, solve for the fraction, and then match it with one of the answer options.

Given Data / Assumptions:

• Let the fraction be f.
• The condition is 2f + 1/f = 17/6.
• The fraction is in lowest terms.
• The numerator of the reduced fraction is 3.
• The options list several candidate fractions; exactly one satisfies all conditions.

Concept / Approach:
First, express the fraction as f = 3/k, because the numerator is 3 in lowest terms and k is a positive integer not divisible by 3. Substitute this form into the equation 2f + 1/f = 17/6. This will give an equation in k. Solving this equation will yield the denominator k and therefore the exact fraction. Finally, check which option matches this fraction and verify that it satisfies the original condition.

Step-by-Step Solution:
Let the fraction be f = 3/k, where k is a positive integer and gcd(3, k) = 1. Substitute into 2f + 1/f = 17/6 to get 2*(3/k) + 1/(3/k) = 17/6. Simplify: 6/k + k/3 = 17/6. Multiply throughout by 6k to clear denominators: 6k*(6/k) + 6k*(k/3) = 6k*(17/6). This becomes 36 + 2k^2 = 17k. Rearrange to standard quadratic form: 2k^2 - 17k + 36 = 0. Solve the quadratic: the discriminant is 17^2 - 4*2*36 = 289 - 288 = 1. Thus k = (17 ± 1)/4, giving k = 18/4 = 4.5 or k = 16/4 = 4. Only k = 4 is an integer, so k = 4. Therefore f = 3/4.
Verification / Alternative check:
Check the fraction f = 3/4 directly in the original condition. Compute twice the fraction: 2*(3/4) = 3/2. The reciprocal is 4/3. Their sum is 3/2 + 4/3. Using a common denominator 6, this becomes 9/6 + 8/6 = 17/6, which matches the given condition exactly. None of the other options with numerator 3, such as 3/5, will produce a sum of 17/6 when used in the same way.

Why Other Options Are Wrong:
Option b (2/3) has numerator 2, not 3, so it violates the numerator condition. Options c (4/5) and d (5/4) also do not have numerator 3. Option e (3/5) has the correct numerator but fails the equation: 2*(3/5) + 5/3 does not simplify to 17/6. Therefore, these choices do not satisfy both the algebraic condition and the lowest terms numerator requirement.

Common Pitfalls:
A common mistake is to ignore the numerator condition and simply solve 2f + 1/f = 17/6 directly, which may lead to checking many fractions unnecessarily. Another pitfall is mishandling the algebra when clearing denominators, especially in the step that multiplies through by 6k. Carefully setting up the quadratic equation and verifying that k is an integer ensures you find the correct fraction efficiently.

Final Answer:
The fraction whose double plus its reciprocal equals 17/6 and whose numerator in lowest terms is 3 is 3/4.