A can do a piece of work in 15 days and B can do it in $$22\frac{1}{2}$$ days. They worked together for 6 days and the remaining work was completed by C alone in 6 days. A, B and C together will do the same work in:

Let's assume the total work is 90 units.

A can do a piece of work in 15 days.

The efficiency of A = $$\frac{90}{15}$$ = 6 units/day

B can do it in $$22\frac{1}{2}$$ days.

The efficiency of B = $$\frac{90}{45}\times\ 2$$ = 4 units/day

Let's assume the efficiency of C is 'y'.

They worked together for 6 days and the remaining work was completed by C alone in 6 days.

$$6\times(6+4)+6y = 90$$

$$(6+4)+y=15$$

10+y = 15

y = 15-10 = 5 units/day

Time taken by A, B and C together to do the same work = $$\frac{90}{6+4+5}$$

= $$\frac{90}{15}$$

= 6 days