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A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $$\mathrm{m}$$ is equal to:

JEE · Math · previous-year question

  1. A.125
  2. B.160
  3. C.150correct
  4. D.180

Answer

C. 150

Explanation

To determine the value of $$\mathrm{m}$$, we need to formulate the problem using some basic concepts of arithmetic progression and work. Let's first understand the nature of the problem: Initially, there are $$\mathrm{m}$$ computers, and it is estimated that with these $$\mathrm{m}$$ computers, the assignment can be completed in 17 days. However, due to the crash of 4 computers every day starting from the second day onward, the total time taken extends by 8 days, making it 25 days in total. To begin with, let's define the total work (W) in terms of the number of computers and days: The total work (W) is given by: The amount of work completed each day with $$\mathrm{m}$$ computers for 17 days: $$W = 17m$$ When computers crash, the number of working computers each day forms an arithmetic sequence. On the first day, there are $$\mathrm{m}$$ computers. On the second day, there are $$\mathrm{m} - 4$$ computers, on the third day, there are $$\mathrm{m} - 8$$ computers, and so on. We need to sum this series until 25 days are completed. This can be formulated as: Total work done over 25 days with decrement in the number of computers: $$W = m + (m - 4) + (m - 8) + \ldots + \left[m - 4 \times (n - 1)\right]$$ where $$n$$ is the number of days. Here, $$n = 25$$. Notice that we form an arithmetic series where the first term (a) is $$\mathrm{m}$$ and the common difference (d) is -4. The sum of the first n terms of an arithmetic series is: $$S_n = \frac{n}{2} \left[ 2a + (n - 1)d \right]$$ Plugging in the values: $$S_{25} = \frac{25}{2} \left[ 2m + (25 - 1)(-4) \right]$$ $$S_{25} = \frac{25}{2} \left[ 2m - 96 \right]$$ $$S_{25} = \frac{25}{2} \left[ 2m - 96 \right] = 25(m - 48)$$ This work should be equivalent to the work calculated earlier, so: $$17m = 25(m - 48)$$ Solving for $$\mathrm{m}$$: $$17m = 25m - 1200$$ $$8m = 1200$$ $$m = 150$$ Thus, the value of $$\mathrm{m}$$ is equal to: Option C: 150

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