A is three times more efficient than B. When work- ing together, they can complete a work in 24 days. Find the time taken by each to complete the work individually.

Let's denote the time taken by B to complete the work individually as \( x \) days.

Since A is three times more efficient than B, the time taken by A to complete the work individually will be \( \frac{x}{3} \) days.

Now, the rate of work (or efficiency) is the reciprocal of the time taken:

- Rate of A: \( \frac{1}{\frac{x}{3}} = \frac{3}{x} \) (fraction of work done by A in one day)
- Rate of B: \( \frac{1}{x} \) (fraction of work done by B in one day)

When A and B work together, their combined rate of work is the sum of their individual rates:

\[
\frac{3}{x} + \frac{1}{x} = \frac{4}{x}
\]

We know they can complete the work together in 24 days, so their combined rate is also \( \frac{1}{24} \):

\[
\frac{4}{x} = \frac{1}{24}
\]

Now, solve for \( x \):

\[
4 \times 24 = x
\]

\[
x = 96
\]

So, B takes 96 days to complete the work individually.

Since A is three times more efficient, A will take:

\[
\frac{96}{3} = 32 \text{ days}
\]

### Summary:
- **A** takes **32 days** to complete the work individually.
- **B** takes **96 days** to complete the work individually.

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. What if A was only twice as efficient as B? How would that change the result?
2. How would the time change if a third person, C, who is as efficient as A, joined them?
3. If A and B worked alternately day by day, how many days would they take to complete the work?
4. What would be the combined rate of work if A and B worked together at half their efficiency?
5. How would the total time change if A’s efficiency increased by 50%?

**Tip:** When dealing with problems involving work and rates, always consider the rate as the reciprocal of the time taken. This approach simplifies combining and comparing efficiencies.

Understand how to solve a work efficiency problem involving two workers, A and B, who can complete a task together in 24 days. Discover their individual work rates and time required for each to complete the task alone. Suitable for high school students learning about rates of work and linear equations.

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