M is thrice as efficient a workman as N, and together M and N finish a piece of work in 30 days. In how many days will M alone be able to complete the entire work?

Introduction / Context:
This question deals with comparative efficiency in time and work. The statement that M is thrice as good a workman as N means that M works three times faster than N. Using their combined time to complete the job, we can infer their individual work rates and then determine how long M alone would take to finish the full work.

Given Data / Assumptions:
- M and N together complete the work in 30 days.
- M is three times as efficient as N.
- Total work is taken as 1 unit.
- Work rates are constant and add linearly when they work together.

Concept / Approach:
Let the daily work rate of N be x units. Then the rate of M is 3x units per day because M is thrice as efficient. Together, their daily rate is 4x. Since they finish the work in 30 days, their combined rate is also equal to 1 / 30 units per day. Equating these two expressions for the combined rate allows us to solve for x and then for M’s individual time to finish the job alone.

Step-by-Step Solution:
Step 1: Let the rate of N = x units per day. Step 2: Then the rate of M = 3x units per day. Step 3: Combined rate of M and N = 3x + x = 4x units per day. Step 4: They finish the whole work in 30 days, so 4x = 1 / 30. Step 5: Solve for x: x = 1 / (30 * 4) = 1 / 120. Step 6: Rate of M = 3x = 3 * (1 / 120) = 1 / 40 units per day. Step 7: Time taken by M alone for the whole work = 1 / (1 / 40) = 40 days.

Verification / Alternative check:
If M alone takes 40 days, then in one day M does 1 / 40 of the work. If N’s rate is 1 / 120, then together they do 1 / 40 + 1 / 120 = 3 / 120 + 1 / 120 = 4 / 120 = 1 / 30 of the work per day, which matches the given combined time of 30 days. This confirms that the calculation is consistent.

Why Other Options Are Wrong:
- 50 and 60 days: These values imply slower rates for M and would lead to a combined rate smaller than 1 / 30, contradicting the given data.
- 45 days: This would give M a rate of 1 / 45, making the combined rate less than 1 / 30 when added to N’s inferred rate.
- 30 days: This would imply M and N have equal efficiency, which contradicts the condition that M is thrice as efficient as N.

Common Pitfalls:
A common mistake is to treat the ratio of efficiency as the ratio of time directly. Since efficiency and time are inversely related, if M is three times as efficient as N, M will take one third of the time N takes, not three times. Another error is to forget to equate the algebraic expression for the combined rate (4x) with the numerical value (1 / 30) computed from their joint time. Always keep the relationship between rate, time, and work clear.

Final Answer:
M alone will finish the entire work in 40 days.