A and B started working on a project and finished it in 5 days. If A had worked twice as efficiently and B worked at only half of his efficiency then the same job could have been competed in 4 days. A alone can could do it in how many days?

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.

A and B started working on a project and finished it in 5 days. If A had worked twice as efficiently and B worked at only half of his efficiency then the same job could have been completed in 4 days. A alone can do it in how many days?

Solve a work and time problem where A and B collaborate to finish a project in 5 days. Explore how changing their efficiencies affects the total time, and calculate how long A alone would take to complete the project. This problem involves efficiency, simultaneous equations, and algebra, suitable for Grades 9-12.

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