P is faster than Q. P and Q each walk 24 km. The sum of their speeds is 7 km/h and the sum of the times taken by them is 14 hours. What is the speed of P (in km/h)?

Introduction / Context:
This problem involves two people walking the same distance at different speeds. You are given the sum of their speeds and the sum of the times taken by them and asked to determine the speed of the faster walker. The question tests algebraic setup using time = distance / speed and solving simultaneous equations, which is a very common pattern in aptitude exams.

Given Data / Assumptions:
- Distance walked by P = 24 km.
- Distance walked by Q = 24 km.
- Let speeds of P and Q be v_p and v_q km/h respectively.
- v_p + v_q = 7 km/h.
- Sum of times taken: (24 / v_p) + (24 / v_q) = 14 hours.
- P is faster than Q, so v_p > v_q.

Concept / Approach:
We use the relationship time = distance / speed for each person and combine the information into two equations: one for the sum of speeds and another for the sum of times. Solving this system of equations will give both speeds. Then we use the condition that P is faster to identify P's speed. This is a straightforward application of simultaneous equations in algebra connected with time and distance.

Step-by-Step Solution:
Step 1: Let v_p be the speed of P and v_q be the speed of Q.We are given v_p + v_q = 7. (Equation 1)Step 2: Times taken are 24 / v_p and 24 / v_q hours.Sum of times is 14 hours:24 / v_p + 24 / v_q = 14. (Equation 2)Step 3: Divide Equation 2 by 24 to simplify:1 / v_p + 1 / v_q = 14 / 24 = 7 / 12.Step 4: Use the identity 1 / v_p + 1 / v_q = (v_p + v_q) / (v_p * v_q).So (v_p + v_q) / (v_p * v_q) = 7 / 12.Step 5: Substitute v_p + v_q = 7 into this:7 / (v_p * v_q) = 7 / 12 ⇒ v_p * v_q = 12.Step 6: Now we have a system: v_p + v_q = 7 and v_p * v_q = 12.Step 7: Solve the quadratic t^2 - 7t + 12 = 0 for t, whose roots are v_p and v_q.This factors as (t - 3)(t - 4) = 0, so t = 3 or t = 4.Step 8: Since P is faster, P's speed v_p = 4 km/h and Q's speed v_q = 3 km/h.

Verification / Alternative check:
Check the data with v_p = 4 km/h and v_q = 3 km/h. Sum of speeds = 4 + 3 = 7 km/h, which matches the given condition. Time for P = 24 / 4 = 6 hours. Time for Q = 24 / 3 = 8 hours. Sum of times = 6 + 8 = 14 hours, which also matches the given condition. Therefore the values are correct.

Why Other Options Are Wrong:
- 3 km/h would assign the faster speed incorrectly to P, contradicting the statement that P is faster than Q.
- 5 km/h and 6 km/h do not satisfy both the sum of speeds and the sum of times simultaneously when you attempt to find a partner speed v_q that fits all conditions.

Common Pitfalls:
Some candidates misinterpret the average concepts or attempt to compute an average speed instead of using the sum of speeds and times correctly. Others may mistakenly treat 14 hours as the time taken by one person rather than the sum. Carefully translating the verbal description into precise equations and then using algebraic techniques avoids such errors.

Final Answer:
The speed of P is 4 km/h.