A web developer named David can create a custom software module in 20 days, whereas Emily can complete the exact same module in 30 days. They collaborate on the module together for 4 days, after which Emily leaves the project. The remaining development work is finished by David and Frank working together over the next 5 days. How many days would it take Frank to build the entire software module working completely alone?

The correct answer is 12 days.

Step 1: Determine the daily work rates of David and Emily.

Let the total work required to build the software module be 1 unit.

David can complete the module in 20 days, so David's rate of work per day is:

$\text{David's rate}=\frac{1}{20}$

Emily can complete the module in 30 days, so Emily's rate of work per day is:

$\text{Emily's rate}=\frac{1}{30}$

Step 2: Calculate the work completed by David and Emily together in 4 days.

The combined work rate of David and Emily working together per day is:

$\text{Combined rate}=\frac{1}{20}+\frac{1}{30}=\frac{3+2}{60}=\frac{5}{60}=\frac{1}{12}$

Since they collaborate for 4 days, the work done in these 4 days is:

$\text{Work done in 4 days}=4\times \frac{1}{12}=\frac{4}{12}=\frac{1}{3}$

Step 3: Calculate the remaining work.

Subtract the completed work from the total work:

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

Step 4: Determine how long it would take Frank to complete the entire module alone.

Let $F$ be the number of days it takes Frank to build the entire module working alone. Frank's rate per day is:

$\text{Frank's rate}=\frac{1}{F}$

David and Frank work together for 5 days to finish the remaining $\frac{2}{3}$ of the work:

$5\times \left(\frac{1}{20}+\frac{1}{F}\right)=\frac{2}{3}$

Expanding the equation:

$\frac{5}{20}+\frac{5}{F}=\frac{2}{3}$

Simplifying $\frac{5}{20}$ to $\frac{1}{4}$:

$\frac{1}{4}+\frac{5}{F}=\frac{2}{3}$

Subtracting $\frac{1}{4}$ from both sides:

$\frac{5}{F}=\frac{2}{3}-\frac{1}{4}$

$\frac{5}{F}=\frac{8-3}{12}=\frac{5}{12}$

Solving for $F$:

$F=12$

Therefore, it would take Frank 12 days to build the entire software module working completely alone.