Ravi Sotan, and Tanmay are three gardeneis assigned to complete landscaping work. They can finish the work individually in 6, 18, and 24 days respectively. On any given day, all three do not work together and no gardener can work for more than two consecutive days. What is the minimum number of days in which the work will be completed?

We are given three gardeners: Ravi, Sotan, and Tanmay, who can complete a landscaping job individually in 6, 18, and 24 days respectively. They can work on the task but not for more than two consecutive days each, and they never work together on the same day. The goal is to find the minimum number of days required to finish the work.

### Step-by-Step Approach:

#### Step 1: Calculate their individual work rates
The work rate of each gardener can be found by dividing 1 by the number of days each one takes to finish the job:

- Ravi’s rate = \( \frac{1}{6} \) work/day
- Sotan’s rate = \( \frac{1}{18} \) work/day
- Tanmay’s rate = \( \frac{1}{24} \) work/day

#### Step 2: Total work required
The total work required to complete the landscaping is 1 unit (representing the whole job).

#### Step 3: Calculate how much work they can complete on any day
Let’s calculate how much work they complete when they work on different days. Suppose one possible cycle involves Ravi, Sotan, and Tanmay working consecutively (one after another). The work done in three days (one cycle) is:

\[
\text{Work done in 1 day by Ravi} = \frac{1}{6}
\]
\[
\text{Work done in 1 day by Sotan} = \frac{1}{18}
\]
\[
\text{Work done in 1 day by Tanmay} = \frac{1}{24}
\]

The total work done in a 3-day cycle is:

\[
\text{Total work in 3 days} = \frac{1}{6} + \frac{1}{18} + \frac{1}{24}
\]

We need to find the least common denominator (LCD) of 6, 18, and 24, which is 72:

\[
\frac{1}{6} = \frac{12}{72}, \quad \frac{1}{18} = \frac{4}{72}, \quad \frac{1}{24} = \frac{3}{72}
\]

Thus, the total work done in 3 days is:

\[
\frac{12}{72} + \frac{4}{72} + \frac{3}{72} = \frac{19}{72}
\]

#### Step 4: Calculate how many cycles are needed
Now we need to determine how many such cycles (3-day cycles) are needed to complete the entire job. If \( \frac{19}{72} \) of the work is done in 3 days, then we can find how many cycles it will take to complete 1 unit of work:

\[
\text{Number of cycles} = \frac{1}{\frac{19}{72}} = \frac{72}{19} \approx 3.79
\]

This means it will take a little under 4 cycles to complete the job. Since each cycle is 3 days, we estimate the total number of days to be:

\[
3 \times 4 = 12 \text{ days}
\]

#### Step 5: Calculate the exact number of days
After 3 full cycles (9 days), \( 3 \times \frac{19}{72} = \frac{57}{72} \) of the work will be done. The remaining work is:

\[
1 - \frac{57}{72} = \frac{15}{72} = \frac{5}{24}
\]

We need to distribute the remaining \( \frac{5}{24} \) units of work. If Ravi works next, he can complete \( \frac{1}{6} = \frac{4}{24} \) of the work in 1 day, leaving:

\[
\frac{5}{24} - \frac{4}{24} = \frac{1}{24}
\]

Tanmay can finish the remaining \( \frac{1}{24} \) in 1 more day.

Thus, after 11 days, the entire job will be completed.

### Conclusion:
The minimum number of days required to complete the landscaping work is **11 days**.

Would you like more details on any part of this process or any clarifications?

Here are five related questions to deepen your understanding:
1. How would the solution change if Ravi could only work for one consecutive day?
2. What would happen if all three gardeners worked together on some days?
3. How would we adjust the calculation if one of the gardeners could work an extra day?
4. How would the minimum number of days change if a fourth gardener joined with a different rate?
5. How do variations in individual work rates impact the total time to complete the task?

**Tip:** Breaking down work rates into units of work per day can help solve many time-based work problems efficiently.

Ravi, Sotan, and Tanmay are three gardeners assigned to complete landscaping work. They can finish the work individually in 6, 18, and 24 days respectively. On any given day, all three do not work together and no gardener can work for more than two consecutive days. What is the minimum number of days in which the work will be completed?

This problem involves calculating the minimum number of days required for three gardeners to complete a landscaping task. With individual work rates and constraints of working no more than two consecutive days, the solution involves the concept of work and time, calculating work rates, and using least common denominators. Suitable for students in Grades 9-11.

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