Raja and Kundan together can build a wall in 20 days, Kundan and Mahesh together can build the same wall in 30 days, and Mahesh and Raja together can build it in 24 days. Working at their usual constant rates, in how many days will Raja, Kundan and Mahesh together complete the wall?

Introduction / Context:
This question involves three workers, Raja, Kundan and Mahesh, who can build a wall in pairs at different combined speeds. The task is to determine how long they would take to build the same wall if all three worked together. This type of problem tests your understanding of simultaneous equations using work rates.

Given Data / Assumptions:

Raja and Kundan together can build the wall in 20 days.
Kundan and Mahesh together can build the wall in 30 days.
Mahesh and Raja together can build the wall in 24 days.
All three work at constant individual rates throughout, and the total wall is treated as one unit of work.

Concept / Approach:
Let the daily work rates of Raja, Kundan and Mahesh be r, k and m respectively. The combined work rates for each pair give us three equations. By adding these equations, we can find the sum r + k + m, which is exactly the rate when all three work together. Once that combined rate is known, we invert it to obtain the time required for the three of them to complete the wall together.

Step-by-Step Solution:
Let total work = 1 wall. From Raja and Kundan together: r + k = 1/20. From Kundan and Mahesh together: k + m = 1/30. From Mahesh and Raja together: m + r = 1/24. Add all three equations: (r + k) + (k + m) + (m + r) = 1/20 + 1/30 + 1/24. Left side simplifies to 2(r + k + m). Compute the right side using LCM of 20, 30 and 24 which is 120. 1/20 = 6/120, 1/30 = 4/120, 1/24 = 5/120. Sum = 6/120 + 4/120 + 5/120 = 15/120 = 1/8. So 2(r + k + m) = 1/8, which gives r + k + m = 1/16. Thus, when all three work together, their combined rate is 1/16 of the wall per day. Time taken by Raja, Kundan and Mahesh together = 1 / (1/16) = 16 days.

Verification / Alternative check:
A quick check is to note that if all three together work at a rate of 1/16 per day, they finish the wall in 16 days. Now compare this to the pairwise times of 20, 30 and 24 days; it is reasonable that adding a third worker increases the overall speed enough to reduce the time to a value slightly less than the fastest pair time, which is 20 days. Therefore, 16 days is a consistent and realistic answer.

Why Other Options Are Wrong:
Option 18 days corresponds to a combined rate of 1/18 per day, which does not satisfy the pair equations when decomposed.
Option 20 days is actually the time for Raja and Kundan only, ignoring Mahesh's contribution.
Option 14 days implies an even higher combined rate, inconsistent with the given pairwise speeds.
Option 15 days also does not match the algebraic result of r + k + m = 1/16 and leads to contradictions if checked with the pair equations.

Common Pitfalls:
Students often try to average the pairwise times instead of working with rates and equations. Another common error is forgetting that when you add the three pairwise equations, you get twice the sum of the three individual rates. Carefully handling fractions and using the LCM for denominators reduces mistakes and leads to the correct combined rate.

Final Answer:
Hence, Raja, Kundan and Mahesh together can build the wall in 16 days, so the correct option is 16 days.