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24 boys will complete a piece of work in 18 days. 2 days after they started the work, 8 more boys joined with them. In how many days will all of them together f

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After 2 days boys have 1/9 of a job done, so 8/9 was left. Their speed increased by 1/3, becoming 4/3 of the initial speed of work. Initial group would have finished the job in 16 days, larger group would finish in 3/4 of the time, that is 12 days.
Since it takes 24 boys 18 days to complete all of the work, the total work to be accomplished can be represented as 24 * 18 or 432 person-days.
In the first 2 days, 24 boys will have accomplished 24*2 or 48 person-days of this total amount.
Subtracting these two values gives 384 person-days as the amount of work remaining after two days.
There are 32 boys available to accomplish the remaining 384 person-days worth of work; Dividing the total remaining work of 384 person-days by 32 persons results in 12 days to accomplish the remaining work.
Let me know if this helps!
George T.