Question

A alone can complete a work in 16 days and B alone in 12 days. Starting with A, they work on alternate days. The total work will be completed in.
A.12 days
B.13 days
C.\[13\dfrac{2}{7}\] days
D.None of these

Answer 1

Hint: We will first convert both the works in fractional terms to find the unit of work done. As given that they are working on alternate days, so, the 2-days work can be calculated from here. As we have found the one pair, we can find the 6-pairs by multiplying 6 with the obtained value as in 6 cycles, A on 1st day and B on the next day would complete the work. Now, from this we can find the remaining work. Now, we will calculate the work done on the 13th day by A and then the work done on 14th day by B. The total days can be calculated by adding all the values.

Complete step-by-step answer:
We will first consider the number of days given and convert the number of days in fractional form to find the number of units of work done.
Thus, we get,
A’s 1-day work= \[\dfrac{1}{{16}}\] units
B’s 1-day work= \[\dfrac{1}{{12}}\] units
As its given that they both are working on alternate days so, their 2 days work can be calculated as:
\[ \Rightarrow \dfrac{1}{{16}} + \dfrac{1}{{12}} = \dfrac{7}{{48}}\] units
Thus, we have found the 1-pair of work done that is \[\dfrac{7}{{48}}\] units.
Now, we can find the work done in 6 pairs by multiplying 6 with the 1-pair of work.
Thus, we get,
\[ \Rightarrow 6 \times \left( {\dfrac{7}{{48}}} \right) = \dfrac{{42}}{{48}}\] units.
Thus, the number of units obtained has been the work done in 12 days.
Now, only 6 units of work remains to be done.

Now, we will evaluate the value of the remaining work.
Thus, we get,
\[ \Rightarrow 1 - \dfrac{{42}}{{48}} = \dfrac{6}{{48}}\]
Now, we will find the work done on the 13th day as it’s A’s turn and A does \[\dfrac{1}{{16}}\] of the whole work in 1 day.
Thus, the remaining work is \[48\left( {\dfrac{1}{{16}}} \right) = 3\] units
Now, we will find the number of days 3 units work will take.
As B comes the next day, on the 14th day, we will find the number of units of work.
That is,
\[ \Rightarrow \dfrac{1}{{12}}\left( {48} \right) = 4\] units
As only 3 units of work is required to be completed on the 14th day and B will take \[\dfrac{3}{4}\] of the day to complete the 3 remaining units of work.
Hence, we will find the total days the work takes to get completed.
\[
   \Rightarrow {\text{total days}} = 12 + 1 + \dfrac{3}{4} \\
   \Rightarrow {\text{total days}} = 13\dfrac{3}{4} \\
 \]
Hence, total work is completed in \[13\dfrac{3}{4}\] days.
As there is no option like this then, we will go with option D that is none of these.


Note: We have to convert the number of days in the units of work by writing it in the fractional form. As both the people work alternatively, we have calculated the number of works done in pairs. For the remaining work we will calculate by subtracting the obtained work from 1.