Answer
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Hint: In this question we have to find the number of days in which some amount of work can be completed or vice versa by a particular number of persons. We are given the number of days for completion of work for three persons A, B and C when done separately. First we will find the amount of work done by each person separately in one day or the rate of doing work and then we will find the time taken by all three persons to complete the work when working together, by adding the amount of work done by each one separately in one single day, which will come out to be \[\dfrac{216}{13}\] and by subtracting 12 from it we will calculate the work done by all three guys together. Similarly we will do for 2 people left A and C, and the rest of the work will be done by last person C alone. To calculate the number of days C worked alone we will find the work left for it to do after both B and A have left by using the equation representing the amount of work done by C alone and by putting value of left over work in it by using a unitary method.
Complete step-by-step answer:
In this question we are given the number of days for the completion of some particular work which is done by three persons separately, all alone.
Now first of all we need to calculate the rate of doing work by these three persons when working separately, or we can also call this quantity as work done by three people separately in one day. We can calculate the one day work done by each person separately in the following way by using unitary method, which is,
Suppose the amount of whole work is \[x\] and we are assuming that the amount of work done every day is the same or constant.
So, time taken by A to complete the whole work \[x\] is 36 days.
Therefore work completed by A alone in one day is,
\[{{W}_{A}}=\dfrac{x}{36}\]
Similarly, time taken by B to complete the whole work \[x\] is 54 days.
Therefore work completed by B alone in one day is,
\[{{W}_{B}}=\dfrac{x}{54}\]
At last, the time taken by C to complete the whole work \[x\] is 72 days.
Therefore work completed by C alone in one day is,
\[{{W}_{C}}=\dfrac{x}{72}\]
Now in the question is it given that all three of the persons started working on the same work but this time they are together, so the equation of work done by them in one day is,
\[{{W}_{A+B+C}}=\dfrac{x}{36}+\dfrac{x}{54}+\dfrac{x}{72}\]
\[{{W}_{A+B+C}}=\dfrac{13x}{216}\]
But this form above is not correct to represent the work done by all three in one day, so the correct form is,
\[{{W}_{A+B+C}}=\dfrac{x}{\left( \dfrac{216}{13} \right)}\]
Therefore time taken by all three together to complete the given work \[x\] is \[\dfrac{216}{13}\] days.
Now we are given that 12 days before completion of the work B left the work and was gone, now only A and C are left to do the left over work, we need to find the amount of work left after B has gone for which we have to subtract 12 from \[\dfrac{216}{13}\] and then we will get the number of days in which B worked and if we multiply the whole work with these days we will get the amount of work completed by three persons in these days.
Mathematically,
Number of days B worked with A and C is \[\dfrac{216}{13}-12=\dfrac{216-156}{13}=\dfrac{60}{13}\] days.
Work done by the three persons is,
\[{{W}_{{}^{60}/{}_{13}\left( A+B+C \right)}}=\dfrac{60}{13}\times \dfrac{x}{\left( \dfrac{216}{13} \right)}=\dfrac{60}{13}\times \dfrac{13}{216}\times x\]
\[{{W}_{{}^{60}/{}_{13}\left( A+B+C \right)}}=\dfrac{5x}{18}\]
Now the amount of pending work which is to be done by A and C is,
\[{{W}_{pending\left( A,C \right)}}=x-\dfrac{5x}{18}\]
\[{{W}_{pending}}=\dfrac{18x-5x}{18}\]
\[{{W}_{pending}}=\dfrac{13x}{18}\]
Now we have to find the amount of work done by A and C in one day, which is,
\[{{W}_{A+C}}=\dfrac{x}{36}+\dfrac{x}{72}\]
\[{{W}_{A+C}}=\dfrac{2x+x}{72}\]
\[{{W}_{A+C}}=\dfrac{3x}{72}=\dfrac{x}{24}\]
Now by looking at the above equation we can say that A and C together can complete the whole work \[x\] in 24 days, so time taken to complete the left over work \[{{W}_{pending}}=\dfrac{13x}{18}\] together by A and C is \[\dfrac{24}{x}\times \dfrac{13x}{18}=\dfrac{52}{3}\] days as 1 work is completed in \[\dfrac{24}{x}\] days.
Now we know that the amount of work done by A and C in 1 day is \[\dfrac{x}{24}\] and the time taken by them to complete the pending work \[{{W}_{pending\left( A+C \right)}}=\dfrac{13x}{18}\] is \[\dfrac{52}{3}\] days.
Now we are given that 8 days before completion of the work A left the work and was gone, now only C is left to do the left over work, we need to find the amount of work left after A has also gone for which we have to subtract 8 from \[\dfrac{52}{3}\] and then we will get the number of days in which A worked and if we multiply the whole work with these days then we will get the amount of work completed by A and C together in these days.
Mathematically,
Number of days A worked with C is \[\dfrac{52}{3}-8=\dfrac{52-24}{3}=\dfrac{28}{3}\] days.
Work done by A and C together is, now the original work is pending work to be done by A and C together so,
\[{{W}_{{}^{28}/{}_{3}\left( A+C \right)}}=\dfrac{28}{3}\times \dfrac{x}{24}\]
\[{{W}_{{}^{28}/{}_{3}\left( A+C \right)}}=\dfrac{7x}{18}\]
Now the amount of pending work done by C alone is,
\[{{W}_{pending,C}}=x-\dfrac{5x}{18}-\dfrac{7x}{18}\]
\[{{W}_{pending,C}}=\dfrac{6x}{18}\]
\[{{W}_{pending,C}}=\dfrac{x}{3}\]
Now the amount of work done by C alone in one day is,
\[{{W}_{C}}=\dfrac{x}{72}\]
Now by looking at the above equation we can say that the time taken by C to complete whole work \[x\] is 72 days, so time taken to complete the left over work \[{{W}_{pending,C}}=\dfrac{13x}{54}\] by C alone is \[\dfrac{72}{x}\times \dfrac{x}{3}=24\] days as 1 work is completed in \[\dfrac{72}{x}\] days.
Therefore the number of days C worked alone is 24 days.
Hence the correct answer is,
Option. D. 24
Note: Remember that the ratio of work done in one single day or the rate of doing work by a person shall always be evaluated first then the question shall be solved by using it, the numerator shall be kept as the amount of total work only so then we can say that the denominator is the time taken to complete the work, represented as \[{{W}_{A+B+C}}=\dfrac{x}{\left( \dfrac{216}{13} \right)}\], but when the ratio is simplified regardless of the numerator and denominator the meaning changes to the amount of work done in one single day which is \[{{W}_{A+B+C}}=\dfrac{13x}{216}\], do not get confused in it. Always use the rate of doing work in a unitary method and then compare with the given conditions in the question. Do not forget to subtract the amount of work done by A, B and C together from whole work when evaluating the left over work which is to be done by C, \[{{W}_{pending,C}}=x-\dfrac{5x}{18}-\dfrac{7x}{18}\].
Complete step-by-step answer:
In this question we are given the number of days for the completion of some particular work which is done by three persons separately, all alone.
Now first of all we need to calculate the rate of doing work by these three persons when working separately, or we can also call this quantity as work done by three people separately in one day. We can calculate the one day work done by each person separately in the following way by using unitary method, which is,
Suppose the amount of whole work is \[x\] and we are assuming that the amount of work done every day is the same or constant.
So, time taken by A to complete the whole work \[x\] is 36 days.
Therefore work completed by A alone in one day is,
\[{{W}_{A}}=\dfrac{x}{36}\]
Similarly, time taken by B to complete the whole work \[x\] is 54 days.
Therefore work completed by B alone in one day is,
\[{{W}_{B}}=\dfrac{x}{54}\]
At last, the time taken by C to complete the whole work \[x\] is 72 days.
Therefore work completed by C alone in one day is,
\[{{W}_{C}}=\dfrac{x}{72}\]
Now in the question is it given that all three of the persons started working on the same work but this time they are together, so the equation of work done by them in one day is,
\[{{W}_{A+B+C}}=\dfrac{x}{36}+\dfrac{x}{54}+\dfrac{x}{72}\]
\[{{W}_{A+B+C}}=\dfrac{13x}{216}\]
But this form above is not correct to represent the work done by all three in one day, so the correct form is,
\[{{W}_{A+B+C}}=\dfrac{x}{\left( \dfrac{216}{13} \right)}\]
Therefore time taken by all three together to complete the given work \[x\] is \[\dfrac{216}{13}\] days.
Now we are given that 12 days before completion of the work B left the work and was gone, now only A and C are left to do the left over work, we need to find the amount of work left after B has gone for which we have to subtract 12 from \[\dfrac{216}{13}\] and then we will get the number of days in which B worked and if we multiply the whole work with these days we will get the amount of work completed by three persons in these days.
Mathematically,
Number of days B worked with A and C is \[\dfrac{216}{13}-12=\dfrac{216-156}{13}=\dfrac{60}{13}\] days.
Work done by the three persons is,
\[{{W}_{{}^{60}/{}_{13}\left( A+B+C \right)}}=\dfrac{60}{13}\times \dfrac{x}{\left( \dfrac{216}{13} \right)}=\dfrac{60}{13}\times \dfrac{13}{216}\times x\]
\[{{W}_{{}^{60}/{}_{13}\left( A+B+C \right)}}=\dfrac{5x}{18}\]
Now the amount of pending work which is to be done by A and C is,
\[{{W}_{pending\left( A,C \right)}}=x-\dfrac{5x}{18}\]
\[{{W}_{pending}}=\dfrac{18x-5x}{18}\]
\[{{W}_{pending}}=\dfrac{13x}{18}\]
Now we have to find the amount of work done by A and C in one day, which is,
\[{{W}_{A+C}}=\dfrac{x}{36}+\dfrac{x}{72}\]
\[{{W}_{A+C}}=\dfrac{2x+x}{72}\]
\[{{W}_{A+C}}=\dfrac{3x}{72}=\dfrac{x}{24}\]
Now by looking at the above equation we can say that A and C together can complete the whole work \[x\] in 24 days, so time taken to complete the left over work \[{{W}_{pending}}=\dfrac{13x}{18}\] together by A and C is \[\dfrac{24}{x}\times \dfrac{13x}{18}=\dfrac{52}{3}\] days as 1 work is completed in \[\dfrac{24}{x}\] days.
Now we know that the amount of work done by A and C in 1 day is \[\dfrac{x}{24}\] and the time taken by them to complete the pending work \[{{W}_{pending\left( A+C \right)}}=\dfrac{13x}{18}\] is \[\dfrac{52}{3}\] days.
Now we are given that 8 days before completion of the work A left the work and was gone, now only C is left to do the left over work, we need to find the amount of work left after A has also gone for which we have to subtract 8 from \[\dfrac{52}{3}\] and then we will get the number of days in which A worked and if we multiply the whole work with these days then we will get the amount of work completed by A and C together in these days.
Mathematically,
Number of days A worked with C is \[\dfrac{52}{3}-8=\dfrac{52-24}{3}=\dfrac{28}{3}\] days.
Work done by A and C together is, now the original work is pending work to be done by A and C together so,
\[{{W}_{{}^{28}/{}_{3}\left( A+C \right)}}=\dfrac{28}{3}\times \dfrac{x}{24}\]
\[{{W}_{{}^{28}/{}_{3}\left( A+C \right)}}=\dfrac{7x}{18}\]
Now the amount of pending work done by C alone is,
\[{{W}_{pending,C}}=x-\dfrac{5x}{18}-\dfrac{7x}{18}\]
\[{{W}_{pending,C}}=\dfrac{6x}{18}\]
\[{{W}_{pending,C}}=\dfrac{x}{3}\]
Now the amount of work done by C alone in one day is,
\[{{W}_{C}}=\dfrac{x}{72}\]
Now by looking at the above equation we can say that the time taken by C to complete whole work \[x\] is 72 days, so time taken to complete the left over work \[{{W}_{pending,C}}=\dfrac{13x}{54}\] by C alone is \[\dfrac{72}{x}\times \dfrac{x}{3}=24\] days as 1 work is completed in \[\dfrac{72}{x}\] days.
Therefore the number of days C worked alone is 24 days.
Hence the correct answer is,
Option. D. 24
Note: Remember that the ratio of work done in one single day or the rate of doing work by a person shall always be evaluated first then the question shall be solved by using it, the numerator shall be kept as the amount of total work only so then we can say that the denominator is the time taken to complete the work, represented as \[{{W}_{A+B+C}}=\dfrac{x}{\left( \dfrac{216}{13} \right)}\], but when the ratio is simplified regardless of the numerator and denominator the meaning changes to the amount of work done in one single day which is \[{{W}_{A+B+C}}=\dfrac{13x}{216}\], do not get confused in it. Always use the rate of doing work in a unitary method and then compare with the given conditions in the question. Do not forget to subtract the amount of work done by A, B and C together from whole work when evaluating the left over work which is to be done by C, \[{{W}_{pending,C}}=x-\dfrac{5x}{18}-\dfrac{7x}{18}\].
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