If machine X works alone at its constant rate, it fills a certain production order of bolts in 4 hours....

1. Translate the problem requirements: We need to find what fraction of the total bolts will be produced by machine X when both machines work together simultaneously. This is asking for X's share of the combined output.
2. Establish individual work rates: Determine how much of the order each machine completes per hour when working alone.
3. Calculate proportional contributions: When working together, each machine contributes according to its rate, so we find the ratio of X's rate to the combined rate.
4. Express as fraction of total work: Convert the proportional contribution into the fraction of bolts that machine X will produce.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're really being asked. We have two machines working together to fill a production order of bolts. The question asks: "what fraction of the number of bolts in the order will be made by machine X?"

This means we need to find Machine X's share of the total work when both machines work simultaneously. Think of it like two people working together to complete a task - we want to know what portion of the final result comes from person X's efforts.

Process Skill: TRANSLATE

2. Establish individual work rates

Let's figure out how fast each machine works when operating alone.

Machine X completes the entire order in 4 hours working alone. This means in 1 hour, Machine X completes \(\frac{1}{4}\) of the order.

Machine Y completes the entire order in 6 hours working alone. This means in 1 hour, Machine Y completes \(\frac{1}{6}\) of the order.

Think of it this way: if the job is like painting a fence, Machine X paints \(\frac{1}{4}\) of the fence per hour, while Machine Y paints \(\frac{1}{6}\) of the fence per hour.

Technical notation:
• Machine X rate = \(\frac{1}{4}\) order per hour
• Machine Y rate = \(\frac{1}{6}\) order per hour

3. Calculate proportional contributions

When both machines work together, each contributes according to its individual rate. The faster machine (X) will naturally produce more bolts than the slower machine (Y).

When working simultaneously for the same amount of time, Machine X's contribution compared to Machine Y's contribution will be in the ratio of their rates:

X's rate : Y's rate = \(\frac{1}{4} : \frac{1}{6}\)

To make this easier to work with, let's find a common denominator. The LCD of 4 and 6 is 12:
• \(\frac{1}{4} = \frac{3}{12}\)
• \(\frac{1}{6} = \frac{2}{12}\)

So the ratio is \(\frac{3}{12} : \frac{2}{12}\), which simplifies to \(3 : 2\)

This means for every 3 bolts that Machine X produces, Machine Y produces 2 bolts.

4. Express as fraction of total work

Now we can find what fraction of the total bolts will be made by Machine X.

If Machine X produces 3 parts and Machine Y produces 2 parts, then:
• Total parts produced = \(3 + 2 = 5\) parts
• Machine X's share = 3 parts out of 5 total parts = \(\frac{3}{5}\)

Let's verify this makes sense: Machine X is faster (completes job in 4 hours vs 6 hours), so it should produce more than half the bolts. Indeed, \(\frac{3}{5} = 0.6\), which is more than half.

4. Final Answer

Machine X will produce \(\frac{3}{5}\) of the total number of bolts in the order.

Checking against the answer choices: This matches choice D: \(\frac{3}{5}\)

Answer: D

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the question is asking for
Students often confuse this question with a "time to complete" problem rather than a "work distribution" problem. They might try to find how long it takes both machines working together to complete the order, instead of finding what fraction of the bolts Machine X produces. The question specifically asks "what fraction of the number of bolts in the order will be made by machine X?" - this is asking for work distribution, not completion time.

2. Incorrectly setting up the work rate relationship
Some students struggle with the concept that when machines work simultaneously, their individual rates determine their proportional contributions. They might think that since the machines work together, they somehow "share" the work equally (50-50), missing that the faster machine naturally produces more bolts in the same time period.

Errors while executing the approach

1. Arithmetic errors when finding common denominators
When converting the rates \(\frac{1}{4}\) and \(\frac{1}{6}\) to equivalent fractions with a common denominator, students often make mistakes. They might incorrectly convert to \(\frac{2}{12}\) and \(\frac{3}{12}\) (reversing the values) instead of the correct \(\frac{3}{12}\) and \(\frac{2}{12}\), which would lead to the wrong ratio and ultimately the wrong answer.

2. Confusing the ratio direction
After correctly finding that the rate ratio is \(3:2\), some students get confused about which number corresponds to which machine. Since Machine X is faster (4 hours vs 6 hours), it should get the larger number (3), but students might accidentally assign the smaller number (2) to Machine X, leading to answer \(\frac{2}{5}\) instead of \(\frac{3}{5}\).

Errors while selecting the answer

1. Selecting the complementary fraction
After correctly calculating that Machine X produces \(\frac{3}{5}\) of the bolts, some students might mistakenly select \(\frac{2}{5}\) (choice B) thinking about Machine Y's contribution instead of Machine X's contribution. This happens when students lose track of which machine the question is asking about in the final step.

Alternate Solutions

Smart Numbers Approach

Key Insight: Instead of working with abstract fractions of the total order, we can assign a specific number of bolts that makes the calculations clean.

Step 1: Choose a smart number for total bolts
Since machine X takes 4 hours and machine Y takes 6 hours, let's set the total number of bolts = 12 (the LCM of 4 and 6). This choice ensures all rates will be whole numbers.

Step 2: Calculate individual rates
• Machine X rate: \(12 \div 4 = 3\) bolts per hour
• Machine Y rate: \(12 \div 6 = 2\) bolts per hour

Step 3: Determine time to complete together
Combined rate = \(3 + 2 = 5\) bolts per hour
Time to complete 12 bolts together = \(12 \div 5 = 2.4\) hours

Step 4: Calculate machine X's contribution
In 2.4 hours, machine X produces: \(3 \times 2.4 = 7.2\) bolts

Step 5: Find the fraction
Fraction made by machine X = \(\frac{7.2}{12} = \frac{72}{120} = \frac{3}{5}\)

Verification: Machine Y produces \(2 \times 2.4 = 4.8\) bolts
Total: \(7.2 + 4.8 = 12\) bolts ✓

The answer is D. \(\frac{3}{5}\)