Question

# If A completes a piece of work in 3 days, which B completes it in 5 days and C takes 10 days to complete the same work, how long will they take to complete the work, if they work together?$(a){\text{ }}\dfrac{{30}}{{19}}{\text{days}} \\ (b){\text{ }}\dfrac{{30}}{{17}}{\text{days}} \\ (c){\text{ }}\dfrac{{40}}{{19}}{\text{days}} \\ (d){\text{ }}\dfrac{{40}}{{17}}{\text{days}} \\$

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Hint: In this one day work of A can be taken out using the unitary method as he completes work in 3 days, similarly one day work of B can be taken out as B completes work in 5 days and the same for C. If they work together then their work in one day can be taken out and thus its reciprocal will be the total days they will take to complete the work.

Given data
A complete piece of work in 3 days.
B complete a piece of work in 5 days.
C completes a piece of work in 10 days.
Therefore,
One day work of A is =$\dfrac{1}{3}$, one day work of B is = $\dfrac{1}{5}$ and one day work of C is =$\dfrac{1}{{10}}$.
So together (A, B and C) one day work is = $\dfrac{1}{3} + \dfrac{1}{5} + \dfrac{1}{{10}}$
Now simplify the above equation we have,
$\Rightarrow \dfrac{1}{3} + \dfrac{1}{5} + \dfrac{1}{{10}} = \dfrac{{10 + 6 + 3}}{{30}} = \dfrac{{19}}{{30}}$
So together (A, B and C) one day work is =$\dfrac{{19}}{{30}}$.
Therefore they take $\dfrac{{30}}{{19}}$ days if they work together.
Hence option (A) is correct.

Note: In these types of work and time problems the unitary method of finding out one day’s work plays a key role. This set of questions has a standard procedure which includes breaking down the problem into a single day entity, and then using questions constraints. No direct formula can’t be used here it is based on solemn general mathematics of unitary method.