Arun, Varun and Tarun, if working alone, can complete a task in 24, 21, and 15 days, respectively. They charge Rs 2160, Rs 2400, and Rs 2160 per day, respectively, even if they are employed for a partial day. On any given day, any of the workers may or may not be employed to work. If the task needs to be completed in 10 days or less, then the minimum possible amount, in rupees, required to be paid for the entire task is?
Show Hint
- \(34400\)
- \(38400\)
- \(47040\)
- \(38880\)
The Correct Option is B
Approach Solution - 1
Step 1: Calculate work rates and cost per unit. Let the total amount of work be the LCM of the individual completion times: \[ \text{LCM}(24,\,21,\,15) = 840 \text{ units}. \] Hence, the daily work rates are: \[ \text{Arun's rate} = \frac{840}{24} = 35 \text{ units/day}, \] \[ \text{Varun's rate} = \frac{840}{21} = 40 \text{ units/day}, \] \[ \text{Tarun's rate} = \frac{840}{15} = 56 \text{ units/day}. \] Their respective daily wages are: \[ \text{Arun: Rs }2160,\quad \text{Varun: Rs }2400,\quad \text{Tarun: Rs }2160. \] Thus, the cost per unit of work is: \[ \text{Arun} = \frac{2160}{35} \approx 61.7 \text{ Rs/unit}, \] \[ \text{Varun} = \frac{2400}{40} = 60 \text{ Rs/unit}, \] \[ \text{Tarun} = \frac{2160}{56} \approx 38.6 \text{ Rs/unit}. \] Therefore, Tarun is the most economical, followed by Varun and then Arun. Step 2: Decide the cost-minimizing approach. The entire work of 840 units must be completed within 10 days. To minimize the total cost, priority should be given to the worker with the lowest cost per unit. Hence, Tarun should be employed to the maximum possible extent, followed by Varun if additional work remains, and Arun only if unavoidable. Step 3: Allocate maximum work to Tarun. In 10 days, Tarun can complete: \[ 56 \times 10 = 560 \text{ units}. \] The cost incurred for Tarun is: \[ 10 \times 2160 = \text{Rs }21600. \] Remaining work after Tarun finishes: \[ 840 - 560 = 280 \text{ units}. \] Step 4: Assign the remaining work to Varun. Varun completes work at the rate of 40 units per day. The number of days required to finish the remaining 280 units is: \[ \frac{280}{40} = 7 \text{ days}. \] This satisfies the 10-day project constraint. The cost for Varun is: \[ 7 \times 2400 = \text{Rs }16800. \] After this, no work remains, and Arun is not required. Step 5: Compute the minimum total cost. \[ \text{Total Cost} = 21600 + 16800 = \text{Rs }38400. \] Hence, the minimum possible amount required to complete the work is Rs \(38400\).
Approach Solution -2
Step 1: Determine work rates and cost efficiency.
Assume the total work is the LCM of their individual times: \[ \text{LCM}(24, 21, 15) = 840 \text{ units.} \] Then their individual rates (in units/day) are:
\[ \text{Arun's rate} = \frac{840}{24} = 35 \text{ units/day}, \] \[ \text{Varun's rate} = \frac{840}{21} = 40 \text{ units/day}, \] \[ \text{Tarun's rate} = \frac{840}{15} = 56 \text{ units/day}. \] Their daily charges are:
\[ \text{Arun: } \text{Rs } 2160 \text{ per day}, \] \[ \text{Varun: } \text{Rs } 2400 \text{ per day}, \] \[ \text{Tarun: } \text{Rs } 2160 \text{ per day}. \] Cost per unit of work:
\[ \text{Arun: } \frac{2160}{35} \approx 61.7 \text{ Rs/unit}, \] \[ \text{Varun: } \frac{2400}{40} = 60 \text{ Rs/unit}, \] \[ \text{Tarun: } \frac{2160}{56} \approx 38.6 \text{ Rs/unit}. \] Thus, Tarun is the cheapest per unit, then Varun, then Arun.
Step 2: Strategy to minimize cost.
We must finish 840 units of work in at most 10 days. To minimize cost:
- Use Tarun as much as possible (most cost-efficient),
- Then use Varun if needed,
- Use Arun only if absolutely necessary.
Step 3: Assign work to Tarun.
Tarun can work at most 10 days (due to the time limit). Work done by Tarun in 10 days: \[ 56 \times 10 = 560 \text{ units.} \] Cost for Tarun: \[ 10 \times 2160 = \text{Rs } 21600. \] Remaining work: \[ 840 - 560 = 280 \text{ units.} \]
Step 4: Assign remaining work to Varun.
Varun's rate is 40 units/day. Days required by Varun to complete 280 units: \[ \frac{280}{40} = 7 \text{ days.} \] This is within the 10-day limit (they can work in parallel or sequentially within 10 days, since the constraint is on total project duration, not individual employment days). Cost for Varun: \[ 7 \times 2400 = \text{Rs } 16800. \] Remaining work: \(0\). Arun is not needed.
Step 5: Total minimum cost. \[ \text{Total Cost} = 21600 + 16800 = \text{Rs } 38400. \] Hence, the minimum possible amount required is Rs \(38400\).
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