Understanding Work Efficiency Relationship

The problem states that M is thrice as efficient as N. Efficiency is the rate at which work is done. If M is thrice as efficient, it means M completes three times the work N does in the same amount of time, or M completes the same work in one-third the time N takes.

Let the time taken by N to finish the job be $D_N$ days.

Let the time taken by M to finish the job be $D_M$ days.

Since M is thrice as efficient as N, the time taken by M will be one-third the time taken by N.

Therefore, $D_M = \frac{D_N}{3}$.

Calculating N's Work Duration

We are given that M finishes the job in 242 days less than N. This can be written as:

$D_N - D_M = 242$ days

Now, substitute the expression for $D_M$ from the efficiency relationship into this equation:

$D_N - \frac{D_N}{3} = 242$

To solve for $D_N$, find a common denominator:

$\frac{3D_N - D_N}{3} = 242$

Combine the terms on the left side:

$\frac{2D_N}{3} = 242$

Isolate $D_N$ by multiplying both sides by 3 and dividing by 2:

$2D_N = 242 \times 3$

$D_N = \frac{242 \times 3}{2}$

$D_N = 121 \times 3$

$D_N = 363$

So, N can finish the entire work in 363 days.

Verification

If N takes 363 days, then M takes $D_M = \frac{363}{3} = 121$ days.

The difference in days is $D_N - D_M = 363 - 121 = 242$ days, which matches the information given in the question.