Question:A can do a piece of work in 10 days which B done can do in 12 days. In how many days will they finish the work, both working together ? 

A's 1 day's work` = 1/10,` B's 1 day's work` = 1/2`

(A + B)'s 1 day's work` = (1/10 + 1/12) = 11/60`.

:. Both will finish the work in` 60/11 = 5 5/11` days.

Question:Two persons A and B working together can dig a trench in 8 hours while A alone can dig it in 12 hours. In how many hours B alone can dig such a trench ? 

(A + B)'s 1 hour' s work `= 1/8` A's 1 hour's work `=1/12`

:. B' s 1 hour's work` = (1/8 - 1/12) = 1/24`

Hence B alone can dig the trench in 24 hours.

Question:A and B can do a piece of work in 12 days ; B and C can do it in 15 days ; A and C  can do it in 20 days. In how many days will A, B and C finish it, working all together ?

Also, find the number of days taken by each to finish it working alone. 

(A + B)'s 1 day's work` = 1/12;` (B + C)' s 1 day's work` = 1/15`

and (A + C)'s 1 day's work` = 1/20`

Adding, we get : 2 (A + B + C)'s 1 day's work

`= (1/12 + 1/15 + 1/20) = 1/5.`

:. (A + B + C)' s 1 day's work` = 1/10`

Thus A,B and C together can finish the work in 10 days.

Now A's 1 day's work

 = [(A + B + C)'s 1 day's work] - [(B + C) 's 1 day's work] 

`= (1/10 - 1/15) = 1/30`

A alone can finish the work in 30 days.

Similarly, B's 1 day's work` = (1/10 - 1/20) = 1/20`

:. B alone can finish the work in 20 days.

And C's 1 day's work` = (1/10 - 1/12) = 1/60.

:. C alone can finish the work in 60 days.

Question:3 men can complete a piece of work in 6 days. Two days after they started the work, 3 more men joined them. How many days will they take to complete the remaining work ? 

Work done by 3 men in 2 days` = (1/6 xx 2) = 1/3`

Remaining work` = (1 - 1/3) = 2/3`

Now 3 men's 1 day's work` = 1/6`

1 men's 1 day's work` = 1/18`

6 men's 1 day's work` = (6 xx 1/18) = 1/3`

Now,` 1/3` work is done by them in 1 day.

`:. 2/3` work is done by them in` (3 xx 2/3) = 2` days.

Question:2 men and 3 boys can do a piece of work in 10 days while 3 men and 2 boys can do the same work in 8 days. In how many days can 2 men and 1 boy do the work ? 

Let 1 man's 1 day's work = x & 1 boy's 1 day's work = y.

Then, `2x + 3y = 1/10` and ` 3x + 2y = 1/8` 

Solving, we get : `x = 7/200` and` y = 1/100.`

:. (2 men + 1 boy)'s 1 day's work

 ` = (2 xx 7/200 + 1 xx 1/100)`

  `= 16/200 = 2/25`

So, 2 men & 1 boy together can finish the work in `25/2 = 12 1/2` days.

Question:A is twice as good a workman as B and together they finish a piece of work in 18 days. In how many days will A alone finish the work ? 

(A's 1 day's work): (B's 1 day's work) = 2 : 1.

(A + B)'s 1 day's work `= 1/18`

Divide` 1/18` in the ratio 2 : 1.

:. A's 1 day's work` = (1/18 xx 2/3) = 1/27`

Hence, A alone can finish the work in 27 days.

Question:A and B undertake to do a piece of work for Rs. 600. A alone can do it in 6 days while B alone can do it in 8 days. With the help of C, they finish it in 3 days. Find the share of each. 

C,s 1 day's work` = 1/3 - (1/6 + 1/8) = 1/24`

:. A : B : C = Ratio of their 1 day's work

`= 1/6 : 1/8 : 1/24 = 4 : 3 : 1`

:. A's share `= Rs. (600 xx 4/8) = Rs. 300`

B's share `= Rs. (600 xx 3/8)`

    `= Rs, 225.`

C,s share = Rs. [600 - (300 + 225)]`

        `= Rs. 75`