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 Option 3: $\frac{150}{7}$ days.

We are given that A completes the whole work in 15 days, and B completes it in 30 days. We need to find how many days C alone would take to complete the whole work.

Step 1: Find the individual work rates per day.

A's work rate = $\frac{1}{15}$ of the work per day
B's work rate = $\frac{1}{30}$ of the work per day

Step 2: Find the combined work rate of A and B working together.

A + B combined rate = $\frac{1}{15}$ + $\frac{1}{30}$ = $\frac{2}{30}$ + $\frac{1}{30}$ = $\frac{3}{30}$ = $\frac{1}{10}$ per day

Step 3: Calculate the work completed by A and B together in 6 days.

Work done in 6 days = 6 × $\frac{1}{10}$ = $\frac{6}{10}$ = $\frac{3}{5}$

Step 4: Find the remaining work after A leaves.

Remaining work = 1 - $\frac{3}{5}$ = $\frac{2}{5}$

Step 5: Find the combined work rate of B and C.

B and C together complete the remaining $\frac{2}{5}$ of the work in 5 days. So their combined rate per day is:

B + C rate = $\frac{\frac{2}{5}}{5}$ = $\frac{2}{25}$ per day

Step 6: Isolate C's individual work rate.

Since B + C rate = $\frac{2}{25}$ and B's rate = $\frac{1}{30}$:

C's rate = $\frac{2}{25}$ - $\frac{1}{30}$

Finding a common denominator (LCM of 25 and 30 is 150):

C's rate = $\frac{12}{150}$ - $\frac{5}{150}$ = $\frac{7}{150}$ per day

Step 7: Find the number of days C alone would take.

If C completes $\frac{7}{150}$ of the work per day, then C alone can finish the whole work in:

Days for C = $\frac{1}{\frac{7}{150}}$ = $\frac{150}{7}$ days

Therefore, C alone can complete the whole work in $\frac{150}{7}$ days.