A and B can do a piece of work in 15 days. while B and C can do the same work in 12 days, and C and A in 10 days. They all work together for 5 days, and then B and C leave. How many days more will A take to finish the work?

Let's assume the total work is 60 units.(LCM of 15, 12 and 10)

A and B can do a piece of work in 15 days.

The efficiency of A and B together = $$\frac{60}{15}$$ = 4 units/day    Eq.(i) 

while B and C can do the same work in 12 days.

The efficiency of B and C together = $$\frac{60}{12}$$ = 5 units/day    Eq.(ii)

C and A can do the same work in 10 days.

The efficiency of C and A together = $$\frac{60}{10}$$ = 6 units/day    Eq.(iii)

So the efficiency of A, B and C together = $$\frac{4+5+6}{2}$$ (By Eq.(i),  Eq.(ii) and Eq.(iii).) 

= $$\frac{15}{2}$$

= 7.5 units/day    Eq.(iv)

From Eq.(ii) and Eq.(iv), we can get the efficiency of A.

the efficiency of A = Eq.(iv)-Eq.(ii)

= 7.5-5

= 2.5

They all work together for 5 days, and then B and C leave.

Work done in 5 days by all three = $$5\times7.5$$ = 37.5

remaining work = 60-37.5

= 22.5

number of more days will A take to finish the remaining work = $$\frac{22.5}{2.5}$$

= 9 days