Directions : Read the given information carefully and answer the questions…
2024
Directions : Read the given information carefully and answer the questions 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 machines cannot process more than one product at a time.
Here is the information about the machines:
Machine A: Takes 10 minutes to process a product. The products enter at 7-minute intervals starting at 9:00 AM.
Machine B: Takes 15 minutes to process a product. It starts working only if 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 must wait until the next half-hour mark.
The factory stops accepting new products for processing at 4:00 PM.
Find the total number of products fully manufactured by 5:00 PM?
Answer: E. None of these — In a strictly serial production line, where every stage handles one unit at a time, the line's total output over any fixed window can never exceed (window…
- A.
38
- B.
39
- C.
37
- D.
40
- E.
None of these
Attempted by 3 students.
Show answer & explanation
Correct answer: E
In a strictly serial production line, where every stage handles one unit at a time, the line's total output over any fixed window can never exceed (window length) ÷ (slowest stage's own repeating cycle time) — no upstream feed rate, queue rule, or downstream detail can push the total above what the single most restrictive stage can physically complete, because every unit must pass through that stage one at a time.
Machine | Stated rule | Effective max cycle (once backlogged) |
|---|---|---|
A | 10 min/unit; fed every 7 min | 10 min/unit (input outruns processing, so A runs back-to-back) |
B | 15 min/unit; starts only after A hands off | 15 min/unit (A feeds every 10 min, faster than B, so B also runs back-to-back) |
C | 20 min processing + a mandatory 5-min cooldown before the next unit | 25 min/unit — fixed, regardless of how fast B feeds it |
D | 12 min processing, but may only START on a half-hour clock mark (:00 or :30) | 30 min/unit once a backlog forms — D can complete at most one unit per half-hour slot |
Machine D's own gate (30 min/unit) is even more restrictive than Machine C's cycle (25 min/unit), so Machine D — not C — sets the true ceiling for the whole line. Tracing the pipeline:
Entries begin at 9:00 AM every 7 minutes; because 7 min < 10 min (A's own time), A immediately backlogs and outputs one unit every 10 minutes: 9:10, 9:20, 9:30, …
B backlogs the same way (10 min feed < 15 min own time) and outputs one unit every 15 minutes, starting 9:25.
C backlogs (15 min feed < 25 min own cycle) and hands a finished unit to D every 25 minutes, first at 9:45.
D can only start a waiting unit at the next half-hour mark. The first unit (ready 9:45) starts at 10:00 and finishes 10:12; because D then idles until its next slot, its actual START times settle into one unit exactly every 30 minutes: 10:00, 10:30, 11:00, …, 4:30 PM (each finishing 12 minutes later).
The slot starting at 4:30 PM finishes at 4:42 PM (within the window); the next slot would start at 5:00 PM and finish at 5:12 PM, after the cutoff. Counting completed slots from 10:00 AM through 4:30 PM (every 30 minutes) gives exactly 14 units fully manufactured by 5:00 PM.
Cross-check with an assumption-free upper bound: ignore Machine D's gate altogether and use only Machine C's stated 25-minute cycle over the full 480-minute (8-hour) working day. Even letting C start instantly at 9:00 AM with zero setup delay, at most 480 ÷ 25 ≈ 19 units could finish processing at C, and each still needs Machine D's minimum 12 minutes afterward — tightening the cap to about 18. So whether one uses Machine D's real 30-minute gate (14) or the most generous possible reading based on Machine C alone (≈18), the achievable total stays far below every one of the four numeric options offered.
Since none of 37, 38, 39, or 40 can be reached under any reasonable reading of the stated machine constraints, none of the numeric options is achievable — the only internally consistent choice is ‘None of these’.