Machine A produces pencils at a constant rate of 9000 pencils per hour, and machine B produces pencils at a...

1. Translate the problem requirements: Machine A produces \(\mathrm{9000\,pencils/hour}\), Machine B produces \(\mathrm{7000\,pencils/hour}\). Together they must produce \(\mathrm{100,000\,pencils}\). Each can operate maximum 8 hours. Find minimum hours for Machine B.
2. Establish the optimization principle: To minimize Machine B's operating time, we need to maximize Machine A's contribution since it's more efficient.
3. Calculate maximum possible contribution from Machine A: Determine how many pencils Machine A can produce in its maximum operating time.
4. Determine minimum required work for Machine B: Calculate remaining pencils needed and convert to minimum operating time for Machine B.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by clearly understanding what we have and what we need to find.

We have two pencil-making machines:

• Machine A makes pencils faster: \(\mathrm{9,000\,pencils\,every\,hour}\)
• Machine B makes pencils slower: \(\mathrm{7,000\,pencils\,every\,hour}\)

Our goal: Together, they must produce exactly \(\mathrm{100,000\,pencils}\)

Our constraints: Each machine can work for at most 8 hours

What we need to find: The minimum number of hours that Machine B must operate

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Establish the optimization principle

Here's the key insight: Since we want Machine B to work as little as possible, we should make Machine A work as much as possible.

Think of it this way - if you had to complete a big project and you had one fast worker and one slow worker, and you wanted the slow worker to do as little work as possible, you'd give the maximum work to the fast worker first.

Machine A is our "fast worker" (\(\mathrm{9,000\,pencils/hour\,vs\,7,000\,pencils/hour}\)), so we should use it to its maximum capacity first.

Process Skill: INFER - Recognizing that minimizing one variable means maximizing the other

3. Calculate maximum possible contribution from Machine A

Since Machine A can work for at most 8 hours, let's see how many pencils it can produce at maximum capacity:

Machine A's maximum production = Rate × Maximum time
Machine A's maximum production = \(\mathrm{9,000\,pencils/hour \times 8\,hours = 72,000\,pencils}\)

So if Machine A works at its maximum (8 hours), it will produce \(\mathrm{72,000\,pencils}\).

4. Determine minimum required work for Machine B

Now we can figure out what's left for Machine B:

Remaining pencils needed = Total required - Machine A's maximum contribution
Remaining pencils needed = \(\mathrm{100,000 - 72,000 = 28,000\,pencils}\)

Since Machine B produces \(\mathrm{7,000\,pencils\,per\,hour}\), the time it needs is:
Machine B's minimum operating time = \(\mathrm{28,000\,pencils \div 7,000\,pencils/hour = 4\,hours}\)

Let's verify: Machine A works 8 hours producing \(\mathrm{72,000\,pencils}\), Machine B works 4 hours producing \(\mathrm{28,000\,pencils}\). Total: \(\mathrm{72,000 + 28,000 = 100,000\,pencils}\) ✓

5. Final Answer

The least amount of time that Machine B must operate is 4 hours.

This matches answer choice A.

Common Faltering Points

Errors while devising the approach

Students often confuse what needs to be minimized. They might think they need to minimize the total time both machines work together, or minimize Machine A's time instead of Machine B's time. This leads them to set up the wrong optimization strategy from the start.

Students frequently miss that "at most 8 hours" is a constraint that limits how much each machine can work. They might assume machines can work unlimited hours and try to solve using only the production requirements, missing the key insight that Machine A's maximum capacity determines the solution.

Many students fail to understand that to minimize Machine B's time, they must maximize Machine A's time. They might try to set up complex equations or systems instead of recognizing this simple complementary relationship.

Errors while executing the approach

Students commonly make mistakes when calculating \(\mathrm{9,000 \times 8 = 72,000}\) or \(\mathrm{28,000 \div 7,000 = 4}\). These seemingly simple multiplications and divisions can lead to wrong final answers even with the correct approach.

After finding Machine A's maximum production, students might subtract incorrectly \(\mathrm{(100,000 - 72,000)}\) or forget this step entirely, trying to work with the full \(\mathrm{100,000}\) requirement for Machine B.

Errors while selecting the answer

If students make calculation errors leading to non-integer results, they might incorrectly convert between decimal and mixed number formats when matching against answer choices, potentially selecting a wrong fractional answer.

Due to confusion about the optimization objective, students who calculated Machine A's time (8 hours) might look for this value among the choices, or if they found Machine B works 4 hours, they might second-guess themselves and select a larger value thinking it should be the "maximum" rather than "minimum."