Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at...

1. Translate the problem requirements: We need to find how long Machine B takes alone to produce 800 nails, given that both machines together take x hours and Machine A alone takes y hours for the same 800 nails.
2. Express rates in terms of work per hour: Convert the given time information into rates (nails per hour) for each machine and their combined operation.
3. Set up the rate relationship equation: Use the fact that when working together, the combined rate equals the sum of individual rates.
4. Solve for Machine B's rate: Isolate Machine B's rate from the equation and convert back to time required.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

• When Machines A and B work together, they produce 800 nails in x hours
• When Machine A works alone, it produces 800 nails in y hours
• We need to find how long Machine B takes alone to produce 800 nails

Think of this like two people working together to complete a job faster than either could alone. We know how long they take together, we know how long one person takes alone, and we want to find how long the other person takes alone.

Process Skill: TRANSLATE - Converting the work scenario into clear mathematical relationships

2. Express rates in terms of work per hour

Instead of thinking about total time, let's think about how much work each machine does per hour:

Machine A's rate: If Machine A produces 800 nails in y hours, then in 1 hour it produces \(\frac{800}{\mathrm{y}}\) nails.

Combined rate: If both machines together produce 800 nails in x hours, then in 1 hour they produce \(\frac{800}{\mathrm{x}}\) nails.

Machine B's rate: Let's call this \(\frac{800}{\mathrm{z}}\) nails per hour, where z is what we're looking for (the hours Machine B needs alone).

This is like saying: "If John can paint a fence in 4 hours, his rate is 1/4 of the fence per hour."

3. Set up the rate relationship equation

Here's the key insight: When two machines work together, their combined rate equals the sum of their individual rates.

In plain English: The amount both machines produce together per hour = amount Machine A produces per hour + amount Machine B produces per hour

Mathematically:

\(\frac{800}{\mathrm{x}} = \frac{800}{\mathrm{y}} + \frac{800}{\mathrm{z}}\)

We can factor out 800:

\(\frac{800}{\mathrm{x}} = 800\left(\frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}\right)\)

Dividing both sides by 800:

\(\frac{1}{\mathrm{x}} = \frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}\)

Process Skill: INFER - Recognizing that combined rates equal the sum of individual rates

4. Solve for Machine B's rate

Now we solve for \(\frac{1}{\mathrm{z}}\) (Machine B's rate), then find z (the time we want):

\(\frac{1}{\mathrm{x}} = \frac{1}{\mathrm{y}} + \frac{1}{\mathrm{z}}\)

Subtract \(\frac{1}{\mathrm{y}}\) from both sides:

\(\frac{1}{\mathrm{z}} = \frac{1}{\mathrm{x}} - \frac{1}{\mathrm{y}}\)

To subtract these fractions, we need a common denominator:

\(\frac{1}{\mathrm{z}} = \frac{\mathrm{y}}{\mathrm{xy}} - \frac{\mathrm{x}}{\mathrm{xy}} = \frac{\mathrm{y}-\mathrm{x}}{\mathrm{xy}}\)

Therefore:

\(\frac{1}{\mathrm{z}} = \frac{\mathrm{y}-\mathrm{x}}{\mathrm{xy}}\)

To find z, we take the reciprocal:

\(\mathrm{z} = \frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\)

But wait! Let's check this against our answer choices. We have \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\), but choice (E) shows \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\). These are the same since y-x = -(x-y).

Process Skill: MANIPULATE - Carefully handling fraction operations and sign conventions

5. Final Answer

Machine B takes \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\) hours to produce 800 nails working alone.

This matches answer choice (E): \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\)

Verification: This makes intuitive sense because:

• If y > x (Machine A is slower than their combined time), then y-x > 0, giving a positive result
• The answer involves both x and y, which we'd expect since we need both pieces of information
• If Machine A were very fast (y very small), then y-x would be negative and large in magnitude, meaning our expression \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\) would be positive (since we're dividing by a negative), which represents Machine B being much slower

Answer: (E) \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\)

Common Faltering Points

Errors while devising the approach

• Misunderstanding what the variables represent: Students often confuse whether x and y represent rates or times. The question states that Machine A produces 800 nails in y hours and both machines together produce 800 nails in x hours - these are times, not rates. Mixing this up leads to setting up completely wrong equations.
• Incorrectly combining work rates: Students might think that when machines work together, their times add up (x = time for A + time for B) rather than understanding that their rates add up. This fundamental misunderstanding of how combined work functions leads to wrong equation setup.
• Setting up the wrong target: Students may try to find Machine B's rate directly instead of finding the time it takes Machine B to complete the same 800 nails. This confusion about what the question is actually asking for can derail the entire solution approach.

Errors while executing the approach

• Fraction arithmetic mistakes: When solving \(\frac{1}{\mathrm{z}} = \frac{1}{\mathrm{x}} - \frac{1}{\mathrm{y}}\), students frequently make errors finding the common denominator or subtracting fractions incorrectly. A common mistake is getting \(\frac{\mathrm{x}-\mathrm{y}}{\mathrm{xy}}\) instead of \(\frac{\mathrm{y}-\mathrm{x}}{\mathrm{xy}}\), which leads to the wrong final answer.
• Reciprocal errors: After finding \(\frac{1}{\mathrm{z}} = \frac{\mathrm{y}-\mathrm{x}}{\mathrm{xy}}\), students sometimes forget to take the reciprocal to find z, or they take the reciprocal incorrectly, leading to answers like \(\frac{\mathrm{y}-\mathrm{x}}{\mathrm{xy}}\) instead of \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\).
• Sign confusion: Students often get confused about whether the answer should be \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\) or \(\frac{\mathrm{xy}}{\mathrm{x}-\mathrm{y}}\), especially when they don't carefully track which variable should be larger based on the problem context.

Errors while selecting the answer

• Not recognizing equivalent expressions: Students might correctly derive \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\) but not realize this is the same as choice (E) \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}\), thinking they made an error because the signs don't match at first glance. They may then incorrectly select choice (D) \(\frac{\mathrm{xy}}{\mathrm{x}-\mathrm{y}}\) thinking it's the same.
• Failing to verify the reasonableness: Students might select an answer that doesn't make logical sense. For instance, if they get a negative time or select an expression that would make Machine B faster than the combined rate, they should recognize this as impossible but often don't check.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart numbers that satisfy the constraint

We need values where x < y (since Machine A is faster than their combined rate). Let's choose:

• x = 2 hours (time for both machines together to produce 800 nails)
• y = 6 hours (time for Machine A alone to produce 800 nails)

These numbers are logical because if Machine A alone takes 6 hours, it makes sense that adding Machine B would reduce the time to 2 hours.

Step 2: Calculate individual rates

• Combined rate of A and B = \(800 ÷ 2 = 400\) nails per hour
• Machine A's rate = \(800 ÷ 6 = \frac{400}{3}\) nails per hour

Step 3: Find Machine B's rate

Machine B's rate = Combined rate - Machine A's rate

Machine B's rate = \(400 - \frac{400}{3} = \frac{1200}{3} - \frac{400}{3} = \frac{800}{3}\) nails per hour

Step 4: Calculate time for Machine B alone

Time for Machine B = \(800 ÷ \frac{800}{3} = 800 × \frac{3}{800} = 3\) hours

Step 5: Verify with answer choices

Let's check each answer choice with x = 2, y = 6:

1. \(\frac{\mathrm{x}}{\mathrm{x}+\mathrm{y}} = \frac{2}{2+6} = \frac{2}{8} = \frac{1}{4} ≠ 3\)
2. \(\frac{\mathrm{y}}{\mathrm{x}+\mathrm{y}} = \frac{6}{2+6} = \frac{6}{8} = \frac{3}{4} ≠ 3\)
3. \(\frac{\mathrm{xy}}{\mathrm{x}+\mathrm{y}} = \frac{2×6}{2+6} = \frac{12}{8} = \frac{3}{2} ≠ 3\)
4. \(\frac{\mathrm{xy}}{\mathrm{x}-\mathrm{y}} = \frac{2×6}{2-6} = \frac{12}{-4} = -3 ≠ 3\)
5. \(\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}} = \frac{2×6}{6-2} = \frac{12}{4} = 3 ✓\)

The answer is (E).