Let the total amount of work be 1 unit.

Case 1: 20% of the work is done by A, and 80% is done by B

• Work done by A: $0.2$
• Work done by B: $0.8$
• Total time to complete the work: 20 days.

Let $A$'s rate of work be $a$ units/day and $B$'s rate of work be $b$ units/day.

From this, the total time can be expressed as: $\frac{0.2}{a} + \frac{0.8}{b} = 20 \quad \text{(1)}$

Case 2: 80% of the work is done by A, and 20% is done by B

• Work done by A: $0.8$
• Work done by B: $0.2$
• Total time to complete the work: 30 days.

From this, the total time can be expressed as: $\frac{0.8}{a} + \frac{0.2}{b} = 30 \quad \text{(2)}$

Solving the system of equations

We now have two equations:

1. $\frac{0.2}{a} + \frac{0.8}{b} = 20$
2. $\frac{0.8}{a} + \frac{0.2}{b} = 30$

Let's solve these equations to find $a$ and $b$.

Multiply equation (1) by 4 and equation (2) by 1 to simplify:

$\frac{0.8}{a} + \frac{3.2}{b} = 80 \quad \text{(3)}$ $\frac{0.8}{a} + \frac{0.2}{b} = 30 \quad \text{(2 repeated)}$

Now subtract equation (2) from equation (3):

$\left( \frac{3.2}{b} - \frac{0.2}{b} \right) = 80 - 30$ $\frac{3.0}{b} = 50$ $b = \frac{3}{50} = \frac{1}{16.67} \, \text{units/day}$

Now, substitute $b = \frac{1}{16.67}$ into equation (1):

$\frac{0.2}{a} + \frac{0.8}{\frac{1}{16.67}} = 20$ $\frac{0.2}{a} + 13.336 = 20$ $\frac{0.2}{a} = 20 - 13.336$ $\frac{0.2}{a} = 6.664$ $a = \frac{0.2}{6.664} = \frac{1}{33.32} \, \text{units/day}$

Working Together

When A and B work together, their combined rate is: $\text{Rate of A + B} = a + b = \frac{1}{33.32} + \frac{1}{16.67}$

Let’s calculate the combined rate and the total number of days required to finish 1 unit of work:

$\frac{1}{T} = \frac{1}{33.32} + \frac{1}{16.67}$

Solving this, we get: $T \approx 11.11 \, \text{days}.$

Thus, if A and B work together, they will complete the work in approximately 11.11 days.

Would you like any further clarification or details?

Here are 5 questions to consider:

1. What is the significance of the rates $a$ and $b$ in this problem?
2. How would the solution change if A and B had different individual rates?
3. Can you derive the solution without solving the system of equations directly?
4. What happens if A and B start working together halfway through the project?
5. How does the combined work rate impact overall efficiency?

Tip: When solving work-rate problems, it's often helpful to represent rates as fractions of the total work and use algebraic methods for more complex situations.