To solve the problem step by step, we can follow these calculations:
### Step 1: Determine the work done by L and M
- L can complete the work in 200 hours. Therefore, the work done by L in one hour is:
\[
\text{Work done by L in 1 hour} = \frac{1}{200}
\]
- M can complete the work in 150 hours. Therefore, the work done by M in one hour is:
\[
\text{Work done by M in 1 hour} = \frac{1}{150}
\]
### Step 2: Determine the work done by L, M, and N together
- L, M, and N together can complete the work in 60 hours. Therefore, the work done by L, M, and N together in one hour is:
\[
\text{Work done by L, M, and N in 1 hour} = \frac{1}{60}
\]
### Step 3: Calculate the combined work done by L and M
- The combined work done by L and M in one hour is:
\[
\text{Combined work of L and M} = \frac{1}{200} + \frac{1}{150}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 200 and 150 is 600. Thus, we convert the fractions:
\[
\frac{1}{200} = \frac{3}{600}, \quad \frac{1}{150} = \frac{4}{600}
\]
Adding these gives:
\[
\text{Combined work of L and M} = \frac{3}{600} + \frac{4}{600} = \frac{7}{600}
\]
### Step 4: Calculate the work done by N
- Now we can find the work done by N alone in one hour by subtracting the combined work of L and M from the total work done by L, M, and N:
\[
\text{Work done by N in 1 hour} = \frac{1}{60} - \frac{7}{600}
\]
To perform this subtraction, we convert \(\frac{1}{60}\) to have a denominator of 600:
\[
\frac{1}{60} = \frac{10}{600}
\]
Thus, we have:
\[
\text{Work done by N in 1 hour} = \frac{10}{600} - \frac{7}{600} = \frac{3}{600} = \frac{1}{200}
\]
### Step 5: Determine how long it takes N to complete the work alone
- Since N can do \(\frac{1}{200}\) of the work in one hour, it will take N:
\[
\text{Time taken by N to complete the work} = 200 \text{ hours}
\]
### Final Answer
N alone can complete the work in **200 hours**.
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