A and B can complete a work in 15 and 30 days, respectively. A and B work together for 6 days, and then A leaves. C joins the work, and B & C together complete the remaining work in 5 days. In how many days can C complete the whole work alone?

The correct answer is:
$\frac{150}{7}$

To find the number of days C takes to complete the entire work alone, we can solve the problem step-by-step using a unit-based approach:

Step 1: Determine the daily work rates of A and B
Let us assume the total work is represented by a common multiple of the days taken by A and B. The Least Common Multiple (LCM) of 15 and 30 is 30 units.
Total Work = 30 units.

Since A can complete the work in 15 days, A's daily work rate (efficiency) is:
$\text{Efficiency of A}=\frac{30}{15}=2\text{ units/day}$
Since B can complete the work in 30 days, B's daily work rate (efficiency) is:
$\text{Efficiency of B}=\frac{30}{30}=1\text{ unit/day}$

Step 2: Calculate the work completed by A and B together in 6 days
The combined daily work rate of A and B is:
$2+1=3\text{ units/day}$
Working together for 6 days, A and B complete:
$3\times 6=18\text{ units}$

Step 3: Calculate the remaining work
The remaining work to be done after A leaves is:
$30-18=12\text{ units}$

Step 4: Find the daily work rate of C
B and C together complete the remaining 12 units of work in 5 days.
Their combined daily work rate is:
$\frac{12}{5}\text{ units/day}$
Since B's daily work rate is 1 unit/day, we can find C's daily work rate (let it be c) as:
$1+c=\frac{12}{5}$
$c=\frac{12}{5}-1$
$c=\frac{7}{5}\text{ units/day}$

Step 5: Calculate the time taken by C to complete the entire work alone
To complete the entire 30 units of work alone at a rate of 7/5 units per day, the time required for C is:
$\text{Time}=\frac{\text{Total Work}}{\text{Efficiency of C}}=\frac{30}{\left(\frac{7}{5}\right)}$
$\text{Time}=\frac{30\times 5}{7}=\frac{150}{7}\text{ days}$

Thus, C can complete the whole work alone in:
$\frac{150}{7}\text{ days}$