Two machines, Y and Z, work at constant rates producing identical items. Machine Y produces 3 items in the same...

1. Translate the problem requirements: Machine Y produces 3 items while Machine Z produces 2 items in the same time period. We need to find how long Z takes to produce the same batch that Y produces in 9 minutes.
2. Establish the rate relationship: Use the given information to determine the relative speeds of the two machines.
3. Apply inverse relationship logic: Since we know the rates are inversely related to time, use this relationship to find Z's time requirement.
4. Calculate and verify: Compute the final answer and check it makes logical sense given the rate difference.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language:

• Machine Y and Machine Z are both making identical items
• In any given time period, Machine Y makes 3 items while Machine Z makes only 2 items
• Machine Y takes 9 minutes to complete a certain batch
• We need to find how long Machine Z takes to make that same batch

Think of it like two friends wrapping presents: if Sarah wraps 3 presents in the time it takes John to wrap 2 presents, and Sarah finishes wrapping a pile of presents in 9 minutes, how long would it take John to wrap that same pile?

Process Skill: TRANSLATE - Converting the machine production scenario into a clear rate comparison problem

2. Establish the rate relationship

Since Machine Y produces 3 items while Machine Z produces 2 items in the same time, we can say:

Machine Y is faster than Machine Z. Specifically, Y's rate compared to Z's rate is 3:2.

This means Machine Y works at \(\frac{3}{2} = 1.5\) times the speed of Machine Z.

Or we can think of it the other way: Machine Z works at \(\frac{2}{3}\) the speed of Machine Y.

Rate relationship: \(\mathrm{Rate\ of\ Y} : \mathrm{Rate\ of\ Z} = 3 : 2\)

3. Apply inverse relationship logic

Here's the key insight: when machines work at different rates, their times are inversely related to their rates.

If Machine Y is 1.5 times faster than Machine Z, then Machine Z will take 1.5 times longer than Machine Y to complete the same job.

Since the rates are in the ratio 3:2, the times will be in the inverse ratio 2:3.

This means: \(\mathrm{Time\ for\ Y} : \mathrm{Time\ for\ Z} = 2 : 3\)

Process Skill: INFER - Recognizing the inverse relationship between rates and times

4. Calculate and verify

Now we can set up the time relationship:

If \(\mathrm{Time\ for\ Y} : \mathrm{Time\ for\ Z} = 2 : 3\), and we know Time for Y = 9 minutes, then:

\(2 : 3 = 9 : \mathrm{Time\ for\ Z}\)

Using the proportion: \(\frac{2}{3} = \frac{9}{\mathrm{Time\ for\ Z}}\)

Cross multiply: \(2 \times (\mathrm{Time\ for\ Z}) = 3 \times 9\)

\(2 \times (\mathrm{Time\ for\ Z}) = 27\)

\(\mathrm{Time\ for\ Z} = \frac{27}{2} = 13.5\) minutes = 13 1/2 minutes

Let's verify this makes sense: Machine Z is slower (produces fewer items per unit time), so it should take longer than 9 minutes. Since Z produces 2/3 as much as Y in the same time, Z should take 3/2 times as long: \(9 \times \frac{3}{2} = 13.5\) minutes ✓

Final Answer

Machine Z takes 13 1/2 minutes to produce the same batch.

The answer is E. 13 1/2

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the rate relationship direction

Students often confuse which machine is faster. Since Machine Y produces 3 items while Z produces 2 items in the same time, some students might incorrectly think Z is faster because they focus on the numbers without understanding that "3 vs 2" means Y produces MORE in the same time period.

2. Not recognizing the inverse relationship between rates and times

Many students struggle with the key concept that when rates are in ratio 3:2, the times will be in the inverse ratio 2:3. They might incorrectly assume that if Y is faster, it should also take proportionally more time, missing the inverse relationship entirely.

3. Setting up incorrect proportions

Students may set up the proportion incorrectly by putting rates and times in the same ratio rather than recognizing they are inversely related. For example, they might write "Rate Y : Rate Z = Time Y : Time Z" instead of "Rate Y : Rate Z = Time Z : Time Y".

Errors while executing the approach

1. Arithmetic errors in cross multiplication

When solving \(\frac{2}{3} = \frac{9}{\mathrm{Time\ for\ Z}}\), students may make calculation errors such as: \(2 \times 9 = 3 \times (\mathrm{Time\ for\ Z})\), getting \(18 = 3 \times (\mathrm{Time\ for\ Z})\), leading to Time for Z = 6 minutes instead of the correct \(\frac{27}{2} = 13.5\) minutes.

2. Incorrect fraction conversion

Students might correctly calculate \(\frac{27}{2}\) but then make errors converting this to a mixed number, potentially getting 13 2/3 or other incorrect conversions instead of 13 1/2.

3. Using the wrong ratio values in calculations

Some students might accidentally use the rate ratio (3:2) directly for time calculations, computing \(9 \times \frac{2}{3} = 6\) minutes instead of using the inverse relationship and calculating \(9 \times \frac{3}{2} = 13.5\) minutes.

Errors while selecting the answer

1. Selecting the decimal answer instead of mixed number

Students who correctly calculate 13.5 minutes might look for "13.5" in the answer choices and not recognize that choice E "13 1/2" is the same value expressed as a mixed number.

2. Choosing an answer that seems "reasonable" without verification

Some students might select choice D (12 minutes) because it seems like a reasonable amount of time longer than 9 minutes, without actually completing their calculations or verification step.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for the batch size

Since Machine Y produces 3 items while Machine Z produces 2 items in the same time period, let's choose a batch size that's convenient to work with. We'll use 6 items as our batch size because 6 is divisible by both 3 and 2, making our calculations clean.

Step 2: Determine each machine's rate

If the batch contains 6 items and Machine Y takes 9 minutes to produce this batch:

• Machine Y's rate = \(6 \text{ items} \div 9 \text{ minutes} = \frac{2}{3} \text{ items per minute}\)

Since Y produces 3 items while Z produces 2 items in the same time:

• When Y produces 6 items, Z produces \(\frac{2}{3} \times 6 = 4\) items in the same time
• Since Y takes 9 minutes to produce 6 items, Z produces 4 items in 9 minutes
• Machine Z's rate = \(4 \text{ items} \div 9 \text{ minutes} = \frac{4}{9} \text{ items per minute}\)

Step 3: Calculate time for Z to produce the same batch

For Machine Z to produce 6 items at a rate of \(\frac{4}{9}\) items per minute:

Time = \(6 \div \frac{4}{9} = 6 \times \frac{9}{4} = \frac{54}{4} = 13.5\) minutes

Step 4: Verify our answer

Machine Z takes 13.5 minutes = 13½ minutes to produce the same 6-item batch that Machine Y produces in 9 minutes. This makes sense because Z is slower than Y (produces fewer items in the same time), so Z should take longer.