Question

Arun, Tarun and Varun work for 24, 21 and 15 days respectively and get paid 2160, 2400 and 2160 rupees respectively. They get paid the same even if they work for a partial day. If the work has to be completed within 10 days or less, what is the minimum amount that has to be paid to complete the entire task?

Hint:

In work-wage problems, always find the 'cost per unit of work' for each worker to determine the most economical choice for completing the task.

Answer 1

Correct Answer: 2160

Step 1: Understanding the Question:
We need to find the minimum cost to complete a task within 10 days using three workers with different work rates and pay rates. The key is to find the most cost-effective worker(s) and check if they can complete the job within the given timeframe.
Step 2: Key Formula or Approach:
1. Define a total amount of work (e.g., by finding the LCM of the days).
2. Calculate each worker's work rate (units of work per day).
3. Calculate each worker's pay rate (rupees per day).
4. Determine the cost per unit of work for each worker to find the most efficient one.
5. Use the most efficient worker(s) to calculate the minimum cost, ensuring the time constraint is met.
Step 3: Detailed Explanation:
Part A: Rates of Work and Pay
- Arun: Works 24 days for \rupee 2160 \(\implies\) Pay rate = \(2160/24 = 90\) rs/day.
- Tarun: Works 21 days for \rupee 2400 \(\implies\) Pay rate = \(2400/21 \approx 114.28\) rs/day.
- Varun: Works 15 days for \rupee 2160 \(\implies\) Pay rate = \(2160/15 = 144\) rs/day.
Part B: Efficiency (Cost per unit of work)
Let the total work be the LCM of 24, 21, and 15, which is 840 units.
- Arun's work rate = \(840/24 = 35\) units/day. Cost per unit = \(\frac{90 \text{ rs/day}}{35 \text{ units/day}} = \frac{18}{7}\) rs/unit.
- Tarun's work rate = \(840/21 = 40\) units/day. Cost per unit = \(\frac{2400/21 \text{ rs/day}}{40 \text{ units/day}} = \frac{60}{21} = \frac{20}{7}\) rs/unit.
- Varun's work rate = \(840/15 = 56\) units/day. Cost per unit = \(\frac{144 \text{ rs/day}}{56 \text{ units/day}} = \frac{18}{7}\) rs/unit.
Arun and Varun are the most cost-effective workers (\(18/7\) rs/unit), while Tarun is more expensive (\(20/7\) rs/unit).
Part C: Minimum Cost Calculation
To minimize the cost, we should use only the cheapest workers: Arun and Varun.
Combined work rate of Arun and Varun = \(35 + 56 = 91\) units/day.
Time required for them to complete the 840-unit task = \(\frac{840}{91} \approx 9.23\) days.
Since 9.23 days is less than the 10-day limit, this is a valid strategy.
Minimum cost = Total work \(\times\) Cost per unit of the cheapest workers
\[ \text{Minimum Cost} = 840 \text{ units} \times \frac{18}{7} \frac{\text{rs}}{\text{unit}} = 120 \times 18 = 2160 \text{ rupees} \] Step 4: Final Answer:
The minimum amount that has to be paid is 2160 rupees.