Ravi, Sohan, and Tanmay are three ganderers assigned to complete landscaping work. they can finish the work individually in 6,18, and 24 days respectively. on any giyday, all 3 do not work together and no Gardner can work for more than 2 consecutive days. what is the minimum number of days in which the work will be completed?

We are tasked with finding the minimum number of days to complete the landscaping work when Ravi, Sohan, and Tanmay each work with specific restrictions. Let's break down the problem and solve it step by step.

### Step 1: Work Rate Calculation
First, we calculate the work rate of each gardener in terms of the fraction of the total work they complete in one day:

- Ravi can finish the work in 6 days, so his work rate is \( \frac{1}{6} \) of the work per day.
- Sohan can finish the work in 18 days, so his work rate is \( \frac{1}{18} \) of the work per day.
- Tanmay can finish the work in 24 days, so his work rate is \( \frac{1}{24} \) of the work per day.

### Step 2: Total Work Per Day by the Gardeners
The total work done by all three gardeners if they worked together for a day would be the sum of their individual work rates:

\[
\text{Total work per day} = \frac{1}{6} + \frac{1}{18} + \frac{1}{24}
\]

To compute this:

- First, find the least common denominator (LCD) of 6, 18, and 24, which is 72.
- Convert each fraction to have the denominator of 72:

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

Adding these:

\[
\text{Total work per day} = \frac{12}{72} + \frac{4}{72} + \frac{3}{72} = \frac{19}{72}
\]

Thus, if all three worked together, they would complete \( \frac{19}{72} \) of the work in one day.

### Step 3: Work Pattern and Rest Constraints
The problem specifies that no gardener can work for more than 2 consecutive days, and all three do not work together on any given day. To minimize the total number of days, we need to find an optimal pattern where each gardener works as much as possible without violating the restrictions.

Let’s break it into shifts such that each gardener gets to work in rotation, but no one works for more than 2 consecutive days. A potential strategy might be a cyclic rotation where each gardener works for 2 consecutive days and then rests.

### Step 4: Determining Total Time to Complete the Work
Since we are looking for the minimum time to complete the work and \( \frac{19}{72} \) of the work can be completed in a day when they work optimally, we can estimate the total time as:

\[
\text{Total days} = \frac{1}{\frac{19}{72}} = \frac{72}{19} \approx 3.79
\]

Since we need an integer number of days, rounding up gives us a total of **4 days**.

Thus, the minimum number of days required to complete the work is **4 days**, assuming the gardeners are working optimally within the given constraints.

---

Would you like further details or have any questions?

### Relative Questions:
1. How can we optimize the work distribution to balance the constraints?
2. What if Ravi worked fewer hours to allow others to work more?
3. How does the rest period affect the overall time to complete a project?
4. How can we calculate work completion when more gardeners are added?
5. How would different work rates (e.g., in hours) affect the scheduling?

### Tip:
When solving work rate problems, always sum individual rates to find how quickly the group can complete the task and carefully consider any constraints in scheduling.

Ravi, Sohan, 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 2 consecutive days. What is the minimum number of days in which the work will be completed?

Learn how to calculate the minimum number of days to complete landscaping work by Ravi, Sohan, and Tanmay under specific constraints. Using work rate formulas and optimization techniques, find how fractions and the least common denominator are used to solve this real-world problem. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation