Problem Statement: Two men or 1 woman or 3 boys can do a piece of work in 66 days. In how many days will the same piece of work be done by 1 man, 1 woman, and 1 boy?

Analysis:
To solve this problem, we need to determine the number of days required for 1 man, 1 woman, and 1 boy to complete the work. We are given the information that:
- Two men can complete the work in 66 days.
- 1 woman can complete the work in 66 days.
- 3 boys can complete the work in 66 days.

Solution:
Let's assume that the efficiency of work done by a man, woman, and boy is represented by M, W, and B respectively.

Since two men can complete the work in 66 days, we can write the equation as:
2M = 66 days

Similarly, for 1 woman and 3 boys, we have:
W = 66 days
3B = 66 days

Now, we need to find the equation for 1 man, 1 woman, and 1 boy working together.

Let's assume the time required for them to complete the work is represented by T.

The combined efficiency of 1 man, 1 woman, and 1 boy can be represented as:
M + W + B = 1/T

Since we know the values of M, W, and B in terms of days, we can substitute them in the equation:
(1/2) + (1/66) + (1/3) = 1/T

Simplifying this equation, we get:
(33 + 1 + 22)/66 = 1/T
(56/66) = 1/T
56T = 66
T = 66/56
T = 1.18 days (approx)

Therefore, 1 man, 1 woman, and 1 boy working together can complete the work in approximately 1.18 days.

one man or 2 women or 3 boys can finish a job in 66 days. the formula to use is:R * P * T = QR = rate per person.P = number of personsT = timeQ = quantity of work. set Q = 1 piece of work.set T = 66set P = the number of personsyou will want to solve for R which will give you the rate per person. for the man:R * P * T = Q becomes:R * 1 * 66 = 1divide both sides of this equation by 66 and you get:R * 1 = 1/66divide both sides of this equation by 1 to get:R = 1/66one man can finish 1/66 of the job in 1 day. for the women:R * P * T = Q becomes:R * 2 * 66 = 1divide both sides of this equation by 66 and you get:R * 2 = 1/66divide both sides of this equation by 2 to get:R = 1/132one woman can finish 1/132 of the job in 1 day. for the boys:R * P * T = Q becomes:R * 3 * 66 = 1divide both sides of this equation by 66 and you get:R * 3 = 1/66divide both sides of this equation by 3 to get:R = 1/198one boy can finish 1/198 of the job in 1 day. the question is:how many days will one man, one woman and one boy together take to finish the same work? the rate of one man is 1/66the rate of one woman is 1/132the rate of 1 boy is 1/198 working together their rates are additive. you get: R * P * T = 1Since P is equal to 1 all around, this formula becomes:R * T = 1The R in this case is the overall rate of the 3 people working together.we get:R = (1/66 + 1/132 + 1/198) which is equivalent to:R = (6/396 + 3/396 + 2/396) which is equivalent to:R = 11/396 R * T = 1 becomes:11/396 * T = 1divide both sidesof this eqution by (11/396) and you get:T = 1 / (11/396) = 1 * (396/11) = 396/11 = 36 days. 1 man and 1 woman and 1 boy, working together, can complete the work in 36 days. the man will have completed 1/66 * 36 = 36/66 of the work in 36 days.the woman will have completed 1/132 * 36 = 36/132 of the work in 36 days.the boy will have completed 1/198 * 36 = 36/198 of the work in 36 days. add these up together and you should get 1.36/66 + 36/132 + 56/198 is equivalent to:(36*6 + 36*3 + 36*2) / 396 which is equivalent to:(216 + 108 + 72) / 396 which is equivalent to:396/396 which is equivalent to 1. solution is correct.answer is 36 days.