Key Concepts and Formulas

• Principle of Constant Work: The total amount of work required to complete a task is constant. This can be measured in "resource-time" units (e.g., computer-days).
• Arithmetic Progression (AP): A sequence of numbers such that the difference between consecutive terms is constant. The sum of the first $n$ terms of an AP is given by $S_n = \frac{n}{2}[2a + (n-1)d]$, where $a$ is the first term and $d$ is the common difference.

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Step-by-Step Solution

1. Calculate the Total Work (Initial Plan)

• Explanation: We first determine the total work required for the assignment based on the initial plan, assuming no computer failures. This will be our benchmark for the total work that needs to be accomplished.
• Initial Plan: $m$ computer systems were planned to finish the assignment in $17$ days.
• Calculation: The total work ($W$) is the product of the number of computers and the number of days. $W = (\text{Number of computers}) \times (\text{Number of days})$ $W = m \times 17 \text{ computer-days}$ $W = 17m \text{ computer-days}$
• Reasoning: This step quantifies the total effort needed, which remains constant throughout the problem.

2. Determine the Total Time Taken in the Modified Scenario

• Explanation: The problem states that it took $8$ more days to finish the assignment due to computer crashes. We need to find the total duration of the project in this scenario.
• Calculation: $\text{Total days} = \text{Initial planned days} + \text{Additional days}$ $\text{Total days} = 17 + 8 = 25 \text{ days}$
• Reasoning: This establishes the total number of days over which the work was actually completed, which is crucial for calculating the work done in the modified scenario.

3. Analyze the Number of Working Computers Daily (Modified Scenario)

• Explanation: The number of working computers decreases each day. We need to model this daily change.
• Day 1: $m$ computers are working.
• Day 2: 4 computers crashed at the start of the second day, so $m-4$ computers are working.
• Day 3: Another 4 computers crashed at the start of the third day, so $(m-4)-4 = m-8$ computers are working.
• Pattern: The number of working computers on day $k$ follows the pattern: $m, m-4, m-8, \ldots$. This forms an arithmetic progression.
  • First term ($a$) = $m$
  • Common difference ($d$) = $-4$ (since 4 computers are lost each day)
  • Number of terms ($n$) = $25$ (the total number of days taken)
• Reasoning: Identifying the sequence of working computers as an AP allows us to use the sum formula to calculate the total work done over the 25 days.

4. Calculate the Total Work Done (Modified Scenario)

• Explanation: The total work done in the modified scenario is the sum of the work done by the varying number of computers over the $25$ days. We use the formula for the sum of an arithmetic progression.
• Formula: $S_n = \frac{n}{2}[2a + (n-1)d]$
• Calculation: Substituting the values $n=25$, $a=m$, and $d=-4$: $S_{25} = \frac{25}{2}[2m + (25-1)(-4)]$ $S_{25} = \frac{25}{2}[2m + (24)(-4)]$ $S_{25} = \frac{25}{2}[2m - 96]$ To simplify, we can factor out a 2 from the bracket: $S_{25} = \frac{25}{2} \times 2[m - 48]$ $S_{25} = 25(m - 48) \text{ computer-days}$
• Reasoning: This step calculates the total amount of work accomplished under the actual conditions, by summing the daily contributions of computers.

5. Equate Total Work and Solve for $m$

• Explanation: According to the principle of constant work, the total work required for the assignment is the same in both the planned and the actual scenarios. We set the total work from Step 1 equal to the total work from Step 4.
• Equation: $17m = 25(m - 48)$
• Solving for $m$: $17m = 25m - 25 \times 48$ $17m = 25m - 1200$ Subtract $25m$ from both sides: $17m - 25m = -1200$ $-8m = -1200$ Divide by $-8$: $m = \frac{-1200}{-8}$ $m = 150$
• Reasoning: By equating the two expressions for total work, we form a linear equation that allows us to solve for the unknown variable $m$.

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Common Mistakes & Tips

• Careful with "Start of the day": Ensure you correctly interpret when the reduction in computers occurs. "Start of the second day" means the first day has full capacity, and the reduction applies from the second day onwards.
• Correct AP parameters: Double-check the first term ($a$), common difference ($d$), and the number of terms ($n$) for the arithmetic progression. The number of terms is the total duration of the project in the modified scenario.
• Algebraic Accuracy: Pay close attention to signs and distribution when solving the final equation to avoid calculation errors.

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Summary

The problem involves a scenario where the rate of work changes daily due to computer failures. We first establish the total work required based on the initial plan. Then, we model the actual work done by recognizing that the number of working computers forms an arithmetic progression over the extended project duration. By equating the total work from both scenarios, we derive an equation that allows us to solve for the initial number of computer systems, $m$. The value of $m$ is found to be 150.

The final answer is $\boxed{150}$ which corresponds to option (C).