Time Work Question

Question

If $A$, $B$ and $C$ can do a piece of work in $8$, $10$, $12$ days respectively. They started working together. After working for $1$ day $A$ left the work while after $2$ days $A$ joins & $B$ left the work. After $3$ days $B$ joins and $C$ left the work. There is a holiday after $4$ days and $5th$ day they started working together. In how many days work would be finished?

From the calculations I did I got the answer as $9/37$ but it is not present in the multiple choice questions answer list. So I am a bit confused here if I am missing something which is causing me to get the wrong answer.

Edited :

Here are the Ans to be opted from -

1. $5\frac{9}{37}$
2. $5\frac{8}{37}$
3. $8\frac{111}{120}$
4. $5\frac{8}{120}$

Answer 1

For each day A we add $\frac{15}{120}$ , for B we add $\frac{12}{120}$ and for C we add $\frac{10}{120}$

now we make a little table for the first few days ('x' for working day 'o' for not)

Day: 1 - 2 - 3 -4 - 5 - 6 - 7 ...
A....: x - o - o - x - x - x - x ...
B....: x - x - x - o - o - o - x ...
C....: x - x - x - x - x - x - o ...

and we sum up column 1, then 2, and so on, eventually at the $5$th day we reach$\frac{131}{120}$ and the work is done.

Answer 2

The simplest way is to assume a total work of 120 units (lcm of 8,10,12),, then A works @ 15, B works @12 and C @10 units/day

Day $1: A+B+C$

Day $2: B+C$

Day $3: A+C$

Day $4: A+B$

Day $5$ is a holiday. Work done so far $=3(A+B+C) = 3(15+12+10)=111$

Can you now figure out how much more time $A+B+C$ will need to complete the work ?