In algebra, let the ratios p/q, r/s, and t/u all be equal to √5 (that is, p/q = r/s = t/u = √5). Using this common ratio, find the exact value of the expression (3p^2 + 4r^2 + 5t^2) / (3q^2 + 4s^2 + 5u^2).

Introduction / Context:
This question tests your ability to work with proportional relationships and squared terms. When multiple ratios are equal to the same constant, you can express each numerator in terms of the denominator and the given constant. This leads to significant simplification in rational expressions, a technique often used in algebraic manipulation and in simplifying complex fractions in aptitude exams.

Given Data / Assumptions:
- We are given p/q = r/s = t/u = √5, with all denominators non zero.
- We must evaluate (3p^2 + 4r^2 + 5t^2) / (3q^2 + 4s^2 + 5u^2).
- The common ratio √5 can be used to write each numerator in terms of its denominator.
- All variables are real numbers, and basic algebraic rules apply.

Concept / Approach:
From p/q = √5, we can write p = √5·q and hence p^2 = 5q^2. Similarly, r^2 = 5s^2 and t^2 = 5u^2. Substituting these into the numerator transforms it into a multiple of the denominator. The resulting simplification reveals that the whole fraction is just a constant, independent of the particular values of p, q, r, s, t, and u, as long as the ratio condition holds.

Step-by-Step Solution:
1) Use p/q = √5 to get p = √5·q, so p^2 = (√5)^2·q^2 = 5q^2. 2) Similarly, r/s = √5 gives r^2 = 5s^2, and t/u = √5 gives t^2 = 5u^2. 3) Substitute into the numerator: 3p^2 + 4r^2 + 5t^2 = 3·5q^2 + 4·5s^2 + 5·5u^2. 4) This simplifies to 15q^2 + 20s^2 + 25u^2. 5) Factor out 5: 15q^2 + 20s^2 + 25u^2 = 5(3q^2 + 4s^2 + 5u^2). 6) The denominator is precisely 3q^2 + 4s^2 + 5u^2. 7) Therefore, the fraction is (5(3q^2 + 4s^2 + 5u^2)) / (3q^2 + 4s^2 + 5u^2) = 5.

Verification / Alternative check:
Choose simple numbers that satisfy the ratio condition to verify. For example, let q = s = u = 1 and therefore p = r = t = √5. Then p^2 = r^2 = t^2 = 5. The numerator becomes 3·5 + 4·5 + 5·5 = (3 + 4 + 5)·5 = 12·5 = 60. The denominator becomes 3·1^2 + 4·1^2 + 5·1^2 = 12. Thus the ratio is 60 / 12 = 5, which matches the algebraic result.

Why Other Options Are Wrong:
Option A (25) might arise if someone mistakenly squares the entire fraction instead of each term. Option C (1/5) is the reciprocal of the correct answer and comes from confusing p/q = √5 with q/p = √5. Option D (60) is the numerator from the verification example, forgetting to divide by the denominator. Option E (√5) results from using the ratio only once and not accounting for the squares. Only Option B matches the correctly simplified ratio.

Common Pitfalls:
Students sometimes forget that if p/q = k, then p^2/q^2 = k^2, not k. Others treat each term separately but forget to factor out 5 in the numerator, making cancellation less obvious. It is also easy to confuse p/q with q/p. Writing the expressions clearly and keeping track of squares avoids these issues and makes simplification straightforward.

Final Answer:
Using the common ratio p/q = r/s = t/u = √5 and simplifying the squared terms, the fraction evaluates to 5.