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
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Official Correct Answer: A. First, I know that work rate problems usually involve figuring out how much each person can do in a day and then combining their rates to find out how long it takes together. But in this case, it's a bit more complicated because we have three people with different rates and different daily charges. Plus, the task needs to be done in 10 days or less, and we want the cheapest way possible.
Let me jot down the given information:
- Arun can finish the task alone in 24 days. So his work rate is 1/24 per day. He charges Rs 2160 per day.
- Varun can finish it in 21 days, so his rate is 1/21 per day. He charges Rs 2400 per day.
- Tarun can finish it in 15 days, so his rate is 1/15 per day. He charges Rs 2160 per day.
The task needs to be completed in 10 days or less, and we need to minimize the cost. So, we have to figure out the optimal combination of these three workers to finish the job in 10 days at the lowest possible cost.
Hmm, okay. So, the first thing I think is that to minimize the cost, we should try to use the workers who have a lower cost per day and higher work rates. who that is.
Looking at their rates and charges:
- Arun: 1/24 per day at Rs 2160
- Varun: 1/21 per day at Rs 2400
- Tarun: 1/15 per day at Rs 2160
So, Tarun has the highest work rate (1/15) and the same charge as Arun (Rs 2160). Varun has a higher rate than Arun but charges more. So, maybe Tarun is the most cost-effective in terms of work done per rupee.
But wait, maybe combining some workers can get the job done faster, which might save money even if we have to pay more per day. So, it's a balance between how much work they do and how much they charge.
Let me think about the total work required. Let's assume the total work is 1 task. So, the combined work rate of all three working together would be 1/24 + 1/21 + 1/15. Calculate that.
First, find a common denominator for 24, 21, and 15. The least common multiple (LCM) of these numbers is... , 24 factors into 2^3 * 3, 21 is 3*7, and 15 is 3*5. So, LCM would be 2^3 * 3 * 5 * 7 = 8 * 3 * 5 * 7 = 840. So, the common denominator is 840.
Now, converting each rate:
- Arun: 1/24 = 35/840
- Varun: 1/21 = 40/840
- Tarun: 1/15 = 56/840
Adding them up: 35 + 40 + 56 = 131/840 per day.
So together, they can complete 131/840 of the task each day. To find out how many days they need to finish 1 task, we can do 1 / (131/840) = 840/131 ≈ 6.41 days. So, they can finish it in about 6.41 days if all three work together.
But the question allows up to 10 days, so maybe we don't need all three working every day. Perhaps using a combination where some days have two workers and others have one could be cheaper.
Wait, but if we can finish it in 6.41 days with all three, that's under 10 days. So, maybe that's the way to go. But
If all three work together for 6.41 days, the cost would be (2160 + 2400 + 2160) * 6.41. Calculate that.
First, sum their daily rates: 2160 + 2400 + 2160 = 6720 per day.
Then, 6720 * 6.41 ≈ 6720 * 6 + 6720 * 0.41 ≈ 40320 + 2755.2 ≈ 43075.2. That's about Rs 43,075.20.
But looking at the options, the lowest is Rs 38,400. So, maybe there's a cheaper way.
Alternatively, maybe we can use the workers in a way that minimizes the cost without exceeding 10 days. Perhaps using the most efficient workers for fewer days.
Let me think about the rates and costs per day.
First, Calculating the cost-effectiveness per day, which is cost per unit work.
For Arun: 2160 / (1/24) = 2160 * 24 = 51840 per task.
For Varun: 2400 / (1/21) = 2400 * 21 = 50400 per task.
For Tarun: 2160 / (1/15) = 2160 * 15 = 32400 per task.
So, Tarun is the most cost-effective, followed by Varun, then Arun.
So, to minimize cost, we should use Tarun as much as possible, then Varun, and then Arun if needed.
But we also need to finish the task in 10 days or less. So, how much work Tarun can do in 10 days: 10 * (1/15) = 10/15 = 2/3 of the task. That's not enough. So, we need to bring in someone else.
If we use Tarun for all 10 days, he does 2/3. Then we need 1 - 2/3 = 1/3 more.
Who can do 1/3 of the task? how much time each person needs to do 1/3.