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# A and B together can do a piece of work in 12 days which B and C together do in 16 days. After A has been working at it for 5 days, B for 7 days, C finished it in 13 days. In how much time C alone will do the work?

Last updated date: 17th Sep 2024
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Hint:First we find the one day work for A, B and C.Here a given that the work doing by A and B together and also B and C together so we can split the work of A’s, B’s and C’s Finally, we will find time taken by C to complete the work.

Here, it is given that,
A and B together can do a piece of work in 12 days and B and C together do in 16 days
A works for 5 days, B works for 7 days than C completes the remaining work in 13 days
Let’s solve this question arithmetically:
As A and B can finish the work in 12 days’ time, hence
(A + B)’s one day’s work = $\dfrac{1}{{12}}$
Similarly
(B + C)’s one day’s work = $\dfrac{1}{{16}}$
Now, A’s 5 days’ work + B’s 7 days’ work + C’s 13 days’ work = 1
A’s 5 days’ work + B’s 5 days’ work + B’s 2 days’ work + C’s 2 days’ work + C’s 11 days’ work = 1
Or {(A + B)’s 5 days’ work} + {(B + C)’s 2 days’ work} + C’s 11 days’ work = 1
i.e. $\dfrac{5}{12} + \dfrac{2}{16}$ + C’s 11 days’ work = 1
or C’s 11 days’ work $= 1 - \left( {\dfrac{5}{{12}} + \dfrac{2}{{16}}} \right)$
or C’s 11 days’ work $= 1 - \dfrac{{13}}{{24}}$
or C’s 11 days’ work $= \dfrac{{11}}{{24}}$
or C’s 1 day’s work $= \dfrac{{11}}{{24}} \times \dfrac{1}{{11}} = \dfrac{1}{{24}}$
Therefore, C alone can finish the complete work in 24 days.

Note:Mathematically we say that if A can complete a work in n days, work done by A in $1$ day is $\dfrac{1}{n}$. And if A can complete $\dfrac{1}{n}$ part of the work in $1$ day, then A will complete the work in n days.