Machines X and Y run at different constant rates, and machine X can complete a certain job in 9 hours....

1. Translate the problem requirements: Machine X completes a job in 9 hours total. The actual work sequence is: X works alone for 3 hours, then X and Y work together for 4 more hours to finish. We need to find how long Y alone would take for the entire job.
2. Determine what fraction of work remains after X works alone: Calculate how much of the total job X completes in the first 3 hours working solo.
3. Analyze the combined work phase: Figure out how much work X and Y together accomplish in the final 4 hours, which must equal the remaining fraction of the job.
4. Extract Y's individual work rate: Use the information from the combined phase to determine Y's rate of work per hour, then calculate total time for the complete job.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this problem step by step:

We have two machines, X and Y, that work at different speeds. Think of this like two people painting a house - one might be faster than the other.

Here's what we know:

• Machine X can finish the entire job by itself in 9 hours
• The actual work sequence was: X worked alone for 3 hours first, then both X and Y worked together for 4 more hours to finish the job
• We need to find how long Y would take to do the entire job by itself

The key insight is to think about work in terms of "what fraction of the job gets done each hour." If X can finish the whole job in 9 hours, then X completes \(\frac{1}{9}\) of the job each hour it works.

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

2. Determine what fraction of work remains after X works alone

Now let's figure out how much work X accomplished in those first 3 hours working by itself.

Since X completes \(\frac{1}{9}\) of the job each hour, in 3 hours X completes:

\(3 \text{ hours} \times \frac{1}{9} \text{ job per hour} = \frac{3}{9} = \frac{1}{3}\) of the total job

This means after X works alone for 3 hours, there's still \(\frac{2}{3}\) of the job remaining to be completed.

We can verify this makes sense: if \(\frac{1}{3}\) of the job is done, then \(1 - \frac{1}{3} = \frac{2}{3}\) remains.

3. Analyze the combined work phase

Next, both machines work together for 4 hours to finish the remaining \(\frac{2}{3}\) of the job.

During these 4 hours of combined work:

• X continues working at its rate of \(\frac{1}{9}\) job per hour
• Y works at its unknown rate (let's call this \(\frac{1}{t}\) job per hour, where t is what we're trying to find)
• Together they complete exactly \(\frac{2}{3}\) of the job in 4 hours

So in 4 hours of working together:

X completes: \(4 \times \frac{1}{9} = \frac{4}{9}\) of the job

Y completes: \(4 \times \frac{1}{t} = \frac{4}{t}\) of the job

The total work they do together must equal the remaining \(\frac{2}{3}\):

\(\frac{4}{9} + \frac{4}{t} = \frac{2}{3}\)

4. Extract Y's individual work rate

Now we solve for t (the time Y needs to complete the entire job alone):

\(\frac{4}{9} + \frac{4}{t} = \frac{2}{3}\)

Subtract \(\frac{4}{9}\) from both sides:

\(\frac{4}{t} = \frac{2}{3} - \frac{4}{9}\)

To subtract these fractions, we need a common denominator. The LCD of 3 and 9 is 9:

\(\frac{4}{t} = \frac{6}{9} - \frac{4}{9} = \frac{2}{9}\)

Now we can solve for t:

\(\frac{4}{t} = \frac{2}{9}\)

Cross multiply: \(4 \times 9 = 2 \times t\)

\(36 = 2t\)

\(t = 18\)

Let's verify this makes sense: If Y takes 18 hours to do the whole job alone, then Y works at a rate of \(\frac{1}{18}\) job per hour. In the 4 hours of combined work, Y would complete \(4 \times \frac{1}{18} = \frac{4}{18} = \frac{2}{9}\) of the job, which matches our calculation above.

4. Final Answer

Machine Y, working alone, would take 18 hours to complete the entire job.

This corresponds to answer choice (A) 18.

Verification: Total work done = Work by X alone + Work by X and Y together = \(\frac{1}{3} + \frac{2}{3} = 1\) complete job ✓

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the work sequence

Students often misinterpret the problem setup and think that X and Y work together for the entire 7 hours (3 + 4), rather than understanding that X works alone for 3 hours first, then both machines work together for 4 hours. This leads to setting up the wrong equation from the start.

Faltering Point 2: Confusion about what represents "the job"

Some students get confused about what constitutes "the entire job" versus "remaining work." They might try to set up equations where the 4 hours of combined work represents completing the entire job, rather than just the remaining \(\frac{2}{3}\) of the job after X worked alone.

Errors while executing the approach

Faltering Point 1: Arithmetic errors when working with fractions

Students frequently make mistakes when subtracting fractions like \(\frac{2}{3} - \frac{4}{9}\). They might forget to find the common denominator (9), or incorrectly calculate \(\frac{6}{9} - \frac{4}{9}\), leading to wrong values for Y's work rate and ultimately the wrong final answer.

Faltering Point 2: Cross-multiplication errors

When solving \(\frac{4}{t} = \frac{2}{9}\), students sometimes make errors in cross-multiplication. They might incorrectly write \(4 \times 2 = 9 \times t\) instead of \(4 \times 9 = 2 \times t\), or make basic arithmetic mistakes when calculating \(4 \times 9 = 36\).

Errors while selecting the answer

Faltering Point 1: Selecting Y's rate instead of Y's time

After correctly calculating that Y works at a rate of \(\frac{1}{18}\) jobs per hour, some students mistakenly look for \(\frac{1}{18}\) among the answer choices instead of recognizing that the question asks for the time (18 hours) it would take Y to complete the entire job alone.