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 ...
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Official solution
To determine the value of 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 m computers, and it is estimated that with these 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 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 m computers. On the second day, there are m−4 computers, on the third day, there are 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)+…+[m−4×(n−1)]
where n is the number of days. Here, n=25.
Notice that we form an arithmetic series where the first term (a) is m and the common difference (d) is -4. The sum of the first n terms of an arithmetic series is:
Sn=2n[2a+(n−1)d]
Plugging in the values:
S25=225[2m+(25−1)(−4)]S25=225[2m−96]S25=225[2m−96]=25(m−48)
This work should be equivalent to the work calculated earlier, so:
17m=25(m−48)
Solving for m:
17m=25m−12008m=1200m=150
Thus, the value of m is equal to:
Option C: 150