Directions: Read the given information carefully and answer the question based…

2024

Directions: Read the given information carefully and answer the question based on it.

Four machines (A, B, C, and D) are arranged in a production line to manufacture a specific product. Each machine has distinct operational rules and performs a different function in the manufacturing process. The factory works from 9:00 AM to 5:00 PM every day, and each product must pass through the machines in a specific order to be completed. Each machine has a fixed processing time per product, but a machine cannot process more than one product at a time.

Information about the machines:

  • Machine A: Takes 10 minutes to process a product. Products enter at 7-minute intervals starting at 9:00 AM.

  • Machine B: Takes 15 minutes to process a product. It starts working only after Machine A has completed its task and sends the product forward.

  • Machine C: Takes 20 minutes to process a product. It requires a 5-minute cooldown period before processing the next product.

  • Machine D: Takes 12 minutes to process a product. It can only start processing a product if it receives it before or exactly on the half-hour mark; otherwise the product waits until the next half-hour mark.

The factory stops accepting new products for processing at 4:00 PM.

How many products enter Machine A?

  1. A.

    59

  2. B.

    53

  3. C.

    60

  4. D.

    61

  5. E.

    None of these

Attempted by 1 students.

Show answer & explanation

Correct answer: D

Concept

When items arrive at a fixed time interval across a closed time window, the count is NOT simply (duration ÷ interval). It is a fence-post count: the number of equally spaced arrival points from the start time up to and including the end time equals (total duration ÷ interval) + 1. The "+1" appears because BOTH the first arrival (at the start) and the last arrival (at the end) are arrivals that must be counted — just as a straight fence of length L with posts every d metres has L/d gaps but L/d + 1 posts.

Application

  1. Fix the acceptance window. The first product arrives at 9:00 AM, and the factory stops accepting new products AT 4:00 PM. A product arriving exactly at 4:00 PM is the last one accepted before acceptance stops, so the window is treated as inclusive at both ends: 9:00 AM to 4:00 PM.

  2. Find the window length: from 9:00 AM to 4:00 PM is 7 hours = 7 × 60 = 420 minutes.

  3. Count the gaps between consecutive arrivals: arrivals are 7 minutes apart, so the number of gaps is 420 ÷ 7 = 60.

  4. Apply the fence-post rule: number of arrivals = gaps + 1 = 60 + 1. The +1 is the opening 9:00 AM product, and because 420 is an exact multiple of 7 the 60th gap lands the final arrival precisely at 4:00 PM, which is accepted.

Cross-check

Write out the schedule: 9:00, 9:07, 9:14, …, with the k-th arrival at minute 7k (k = 0, 1, 2, …). The largest k with 7k ≤ 420 is k = 60 (minute 420 = 4:00 PM exactly). Counting k = 0 through k = 60 gives 61 arrival instants — confirming the fence-post result.

Why only Machine A matters

  • The question asks only how many products ENTER Machine A, which depends solely on Machine A's arrival schedule (start time, interval, acceptance cut-off).

  • Machine B's wait-for-A rule, Machine C's cooldown, and Machine D's half-hour gating affect downstream throughput and completion, not how many products arrive at A. They are deliberate distractor detail.

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