Let the work done by A in 1 day be $a$ and by B in 1 day be $b$.

Scenario 1: They finish the project in 5 days

In 1 day, A and B together complete $a + b$ work. Since they finish the project in 5 days, the total work done is: $5(a + b) = 1 \quad \text{(1 unit of work represents the entire project)}$ This gives the equation: $a + b = \frac{1}{5} \quad \text{(Equation 1)}$

Scenario 2: A works twice as efficiently, and B works at half efficiency

In this case, A does $2a$ work per day, and B does $\frac{b}{2}$ work per day. They complete the project in 4 days, so: $4(2a + \frac{b}{2}) = 1$ Simplifying: $2a + \frac{b}{2} = \frac{1}{4} \quad \text{(Equation 2)}$

Solving the equations

From Equation 1: $a + b = \frac{1}{5}$ From Equation 2: $2a + \frac{b}{2} = \frac{1}{4}$ We now solve these equations simultaneously.

Step 1: Multiply Equation 1 by 2 to align terms with Equation 2:

$2a + 2b = \frac{2}{5} \quad \text{(Equation 3)}$

Step 2: Subtract Equation 2 from Equation 3:

$(2a + 2b) - (2a + \frac{b}{2}) = \frac{2}{5} - \frac{1}{4}$ Simplifying: $2b - \frac{b}{2} = \frac{8}{20} - \frac{5}{20} = \frac{3}{20}$ Multiply through by 2 to eliminate the fraction: $4b - b = \frac{6}{20}$ $3b = \frac{3}{20}$ $b = \frac{1}{20}$

Step 3: Substitute $b = \frac{1}{20}$ into Equation 1:

$a + \frac{1}{20} = \frac{1}{5}$ $a = \frac{1}{5} - \frac{1}{20} = \frac{4}{20} - \frac{1}{20} = \frac{3}{20}$

Final Step: Find the time A alone takes to complete the work

If A does $\frac{3}{20}$ of the work per day, the time A alone would take to finish the project is: $\frac{1}{\frac{3}{20}} = \frac{20}{3} = 6 \frac{2}{3} \text{ days}$

Thus, A alone can complete the project in $6 \frac{2}{3}$ days.

Would you like more details, or have any questions?

Here are 5 related questions to expand on this:

1. How many days would B alone take to complete the project?
2. What if both A and B improved their efficiency by 50%? How long would it take them to finish the job?
3. If A had worked thrice as efficiently, how would the total time to complete the project change?
4. What is the work rate of A and B combined in the second scenario where efficiencies are altered?
5. How would you model this situation if A and B’s efficiencies changed continuously over time?

Tip: To solve work problems, always relate efficiency to the total job, and try using simultaneous equations to break down complex scenarios.