Each machine at a toy factory assembles a certain kind of toy at a constant rate of one toy every...

1. Translate the problem requirements: We need to find the percent increase in total toy production when 40% of old machines (rate: 1 toy per 3 minutes) are replaced with new machines (rate: 1 toy per 2 minutes). Calculate production for one hour of work.
2. Set up a concrete scenario: Use a specific number of machines to make calculations manageable and avoid working with percentages throughout the problem.
3. Calculate original production capacity: Determine how many toys all machines produce in one hour at their current rates.
4. Calculate new production capacity: Determine the new total production after replacing 40% of machines with faster ones.
5. Find the percent increase: Compare the new production to the original production to calculate the percentage improvement.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to find. We have a toy factory with machines that currently make toys at one rate, and we're going to replace some of them with faster machines. We need to figure out how much more productive the factory becomes.

Here's what we know:

• Current machines: make 1 toy every 3 minutes
• New machines: make 1 toy every 2 minutes
• We're replacing 40% of the current machines with new ones
• We want to know the percent increase in total production over one hour

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Set up a concrete scenario

Instead of working with percentages and variables throughout, let's use real numbers to make this easier to follow. Let's say the factory has 10 machines total. This makes 40% easy to work with since \(40\% \text{ of } 10 \text{ machines} = 4 \text{ machines}\).

So our scenario is:

• Start with: 10 machines (all old type)
• Replace: 4 machines with new faster ones
• End with: 6 old machines + 4 new machines

This concrete approach will make our calculations much clearer, and the percentage result will be the same regardless of how many machines we assume.

3. Calculate original production capacity

Let's figure out how many toys our 10 original machines make in one hour.

Each old machine makes 1 toy every 3 minutes.
In 60 minutes (1 hour), each old machine makes: \(60 \div 3 = 20\) toys

With 10 old machines total:
Original hourly production = \(10 \text{ machines} \times 20 \text{ toys per machine} = 200 \text{ toys per hour}\)

4. Calculate new production capacity

Now let's calculate production after we replace 4 machines with the faster new ones.

Production from remaining 6 old machines:

• Each still makes 20 toys per hour
• \(6 \text{ machines} \times 20 \text{ toys} = 120 \text{ toys per hour}\)

Production from 4 new machines:

• Each new machine makes 1 toy every 2 minutes
• In 60 minutes: \(60 \div 2 = 30\) toys per machine per hour
• \(4 \text{ machines} \times 30 \text{ toys} = 120 \text{ toys per hour}\)

Total new production = \(120 + 120 = 240\) toys per hour

5. Find the percent increase

Now we can calculate the percent increase in production:

Increase in production = \(240 - 200 = 40\) toys per hour

Percent increase = \((\text{Increase} \div \text{Original}) \times 100\%\)
Percent increase = \((40 \div 200) \times 100\% = 0.20 \times 100\% = 20\%\)

Final Answer

The percent increase in the number of toys assembled in one hour is 20%.

This matches answer choice A. 20%.

To verify: We increased production from 200 to 240 toys per hour, which is indeed a 20% increase.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what gets replaced

Students often misread the problem and think that 40% MORE machines are being added to the factory, rather than understanding that 40% of the EXISTING machines are being replaced. This leads them to calculate as if the total number of machines increases, resulting in a much higher percent increase.

2. Confusing the production rates

The problem gives production rates in different formats - "one toy every 3 minutes" vs "one toy every 2 minutes." Students may incorrectly think the new machines are 3 times faster instead of 1.5 times faster (since \(3 \div 2 = 1.5\)), or they might flip the rates and think old machines are faster than new ones.

3. Setting up the wrong comparison

Students may try to compare the increase in speed of individual machines (from 3 minutes to 2 minutes per toy) rather than understanding they need to compare the total factory output before and after the replacement.

Errors while executing the approach

1. Calculation errors with time conversions

When converting from "toys per 3 minutes" or "toys per 2 minutes" to "toys per hour," students frequently make arithmetic mistakes. For example, they might calculate \(60 \div 3 = 18\) instead of 20, or \(60 \div 2 = 25\) instead of 30.

2. Incorrect percent increase formula

Students often use the wrong formula for percent increase. They might calculate \((240 \div 200) \times 100\% = 120\%\) and think the increase is 120%, or they might use \((40 \div 240) \times 100\%\) instead of \((40 \div 200) \times 100\%\), using the new value as the denominator instead of the original value.

3. Adding production incorrectly

When calculating the new total production, students might forget that only 60% of machines remain old (6 out of 10 in the example) and 40% become new machines. They might accidentally calculate as if all 10 machines became new, or add the old and new production incorrectly.

Errors while selecting the answer

No likely faltering points. The calculation directly yields 20%, which clearly corresponds to choice A without any additional interpretation needed.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient number of machines
Let's assume the factory has 10 machines initially. This makes calculating 40% replacement straightforward:
• Original machines: 10
• Machines to be replaced: \(40\% \text{ of } 10 = 4\) machines
• Machines remaining unchanged: 6 machines

Step 2: Calculate original production rate per hour
Each original machine produces 1 toy every 3 minutes:
• In 60 minutes, each machine produces: \(60 \div 3 = 20\) toys
• Total production with 10 machines: \(10 \times 20 = 200\) toys per hour

Step 3: Calculate new production rate per hour
After replacement:
• 6 unchanged machines: \(6 \times 20 = 120\) toys per hour
• 4 new machines (1 toy every 2 minutes): Each produces \(60 \div 2 = 30\) toys per hour
• 4 new machines total: \(4 \times 30 = 120\) toys per hour
• New total production: \(120 + 120 = 240\) toys per hour

Step 4: Calculate percent increase
Percent increase = \(\frac{\text{New production} - \text{Original production}}{\text{Original production}} \times 100\%\)
= \(\frac{240 - 200}{200} \times 100\%\)
= \(\frac{40}{200} \times 100\%\)
= \(0.20 \times 100\% = 20\%\)

The smart number approach works perfectly here because we can choose any convenient number of machines (10 in this case) and the percentage increase will remain the same regardless of the actual number of machines in the factory.