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?
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?
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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.
Updated On: Jan 2, 2026
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Correct Answer: 2160
Solution and Explanation
Step 1: Understanding the Question
We are required to find the minimum cost to complete a given work within 10 days using three workers who differ in both working efficiency and daily wages. The goal is to identify the most cost-effective worker(s) and ensure the work is completed within the given time limit.
Step 2: Key Approach
- Assume the total work as the LCM of the given days.
- Find each worker’s work rate (units/day).
- Find each worker’s daily wage.
- Compute cost per unit of work.
- Select the cheapest worker(s) satisfying the time constraint.
Step 3: Detailed Solution
Part A: Rates of Work and Pay
- Arun: Completes work in 24 days for ₹2160
Pay rate = \( \frac{2160}{24} = 90 \) rs/day - Tarun: Completes work in 21 days for ₹2400
Pay rate = \( \frac{2400}{21} \approx 114.28 \) rs/day - Varun: Completes work in 15 days for ₹2160
Pay rate = \( \frac{2160}{15} = 144 \) rs/day
Part B: Efficiency (Cost per Unit of Work)
Let total work be the LCM of 24, 21, and 15: \[ \text{Total Work} = 840 \text{ units} \]
- Arun:
Work rate = \( \frac{840}{24} = 35 \) units/day
Cost per unit = \( \frac{90}{35} = \frac{18}{7} \) rs/unit - Tarun:
Work rate = \( \frac{840}{21} = 40 \) units/day
Cost per unit = \( \frac{114.28}{40} = \frac{20}{7} \) rs/unit - Varun:
Work rate = \( \frac{840}{15} = 56 \) units/day
Cost per unit = \( \frac{144}{56} = \frac{18}{7} \) rs/unit
Arun and Varun are the most economical workers, each costing \( \frac{18}{7} \) rs/unit, while Tarun is comparatively expensive.
Part C: Minimum Cost Calculation
To minimize cost, we use only Arun and Varun.
Combined work rate: \[ 35 + 56 = 91 \text{ units/day} \] Time required: \[ \frac{840}{91} \approx 9.23 \text{ days} \] Since this is less than 10 days, the strategy is valid.
Minimum cost: \[ 840 \times \frac{18}{7} = 120 \times 18 = 2160 \]
Step 4: Final Answer
The minimum amount that has to be paid is ₹2160.
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