If 20% of a certain quantity of work is done by A and after that the remaining 80% of the work is done by B, the work will be completed in 20 days. If 80% of the work is done by A and after that the remaining 20% of the work is done by B, the work will be completed in 30 days. If A & B work together how many days are required to complete the work?

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.

Learn how to solve a work rate problem where A and B complete different portions of work at different rates. The problem uses the system of equations to calculate the combined rate and the time required for A and B to complete the work together. Suitable for students in Grades 9-12.

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