To solve this problem, we need to find the ratio of work done by each individual, and then use that ratio to determine the time taken by one man, one woman, and one child to complete the work.

Let's assume that the work done by 1 man in 1 day is M units, the work done by 1 woman in 1 day is W units, and the work done by 1 child in 1 day is C units.

According to the given information, 2 men, or 4 women, or 5 children can complete the work in 38 days. This means that the work done by these groups combined in 38 days is equal to the total work.

So, we can write the following equations:

2M * 38 = Total work
4W * 38 = Total work
5C * 38 = Total work

Now, we need to find the ratios of M, W, and C:

M = (Total work) / (2 * 38)
W = (Total work) / (4 * 38)
C = (Total work) / (5 * 38)

Now, we can find the work done by one man, one woman, and one child in 1 day:

1 man = (Total work) / (2 * 38)
1 woman = (Total work) / (4 * 38)
1 child = (Total work) / (5 * 38)

To find the time taken by one man, one woman, and one child to complete the work, we need to divide the total work by the work done by each individual in 1 day:

Time = Total work / [(Total work) / (2 * 38) + (Total work) / (4 * 38) + (Total work) / (5 * 38)]

Simplifying the expression:

Time = (2 * 38 * 4 * 38 * 5 * 38) / [(2 * 38) + (4 * 38) + (5 * 38)]

The common factor of 38 in the numerator and denominator can be canceled out:

Time = (2 * 4 * 5 * 38) / (2 + 4 + 5)

Time = (40 * 38) / 11

Time = 40 * 3.45

Time ≈ 138

Therefore, 1 man, 1 woman, and 1 child will take approximately 138 days to complete the work.

Since none of the given options match this value, the correct answer is None of these.