In western Europe, x bicycles were sold in each of the years 1990 and 1993. The bicycle producers of Western...

1. Translate the problem requirements: We need to find the decrease in bicycles produced by Western European producers specifically (not total market), given their market share dropped from \(42\%\) to \(33\%\) while total market size remained x bicycles
2. Calculate production volumes for each year: Determine how many bicycles Western European producers made in 1990 versus 1993 using their respective market shares
3. Find the absolute decrease: Calculate the difference between 1990 and 1993 production levels
4. Match with answer choices: Verify our result aligns with the given options

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's actually happening here in plain English. We have a bicycle market in Western Europe where the same number of bicycles (let's call it x) were sold in both 1990 and 1993. However, the share of bicycles that were actually produced by Western European companies changed during this time.

Think of it like this: imagine there's a pie representing all bicycle sales, and that pie stayed the same size (x bicycles) in both years. But the slice of that pie belonging to Western European producers got smaller - it went from \(42\%\) in 1990 down to \(33\%\) in 1993.

We need to find how many fewer bicycles the Western European producers made and sold between these two years.

Process Skill: TRANSLATE - Converting the market share language into a clear mathematical understanding of what we're calculating

2. Calculate production volumes for each year

Now let's figure out exactly how many bicycles Western European producers made in each year.

In 1990: Western European producers had \(42\%\) of the market

• So they produced and sold: \(42\%\) of x bicycles
• In mathematical terms: \(0.42 \times \mathrm{x} = 0.42\mathrm{x}\) bicycles

In 1993: Western European producers had \(33\%\) of the market

• So they produced and sold: \(33\%\) of x bicycles
• In mathematical terms: \(0.33 \times \mathrm{x} = 0.33\mathrm{x}\) bicycles

Notice how the total market size (x) stayed the same, but Western European producers' portion got smaller.

3. Find the absolute decrease

To find the decrease, we need to subtract what they produced in 1993 from what they produced in 1990.

Decrease = Production in 1990 - Production in 1993
Decrease = \(0.42\mathrm{x} - 0.33\mathrm{x}\)
Decrease = \(0.09\mathrm{x}\)

In percentage terms, \(0.09 = 9\%\), so the decrease is \(9\%\) of x.

This makes intuitive sense: they lost 9 percentage points of market share (from \(42\%\) down to \(33\%\)), and since the total market size was x, they lost \(9\%\) of x bicycles.

4. Match with answer choices

Looking at our result of \(0.09\mathrm{x}\), which equals \(9\%\) of x, let's check the answer choices:

1. \(9\%\) of \(\mathrm{x}/100\) - This would be much smaller than our answer
2. \(14\%\) of \(\mathrm{x}/100\) - This would also be much smaller than our answer
3. \(75\%\) of \(\mathrm{x}/100\) - This would also be much smaller than our answer
4. \(9\%\) of x - This matches our answer exactly!
5. \(14\%\) of x - This is larger than our calculated decrease

5. Final Answer

The decrease in the annual number of bicycles produced and sold in Western Europe from 1990 to 1993 is \(9\% \text{ of } \mathrm{x}\).

The correct answer is D) \(9\% \text{ of } \mathrm{x}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "market share" means

Students may confuse market share with actual production numbers. They might think that since the same number of bicycles (x) were sold in both years, Western European producers also made the same amount both years. They fail to recognize that market share represents the percentage of total sales that came from Western European producers, not the absolute production volume.

2. Incorrectly identifying what needs to be calculated

Some students may try to calculate the percentage decrease in market share (which would be something like \((42-33)/42\)) instead of calculating the absolute decrease in the number of bicycles produced. The question asks for the decrease in the "annual number of bicycles," not the percentage change in market share.

3. Misunderstanding the relationship between total market and producer output

Students might think they need to find the change in total market size rather than understanding that the total market stayed constant at x bicycles, and only the Western European producers' portion of that market changed.

Errors while executing the approach

1. Arithmetic errors in percentage calculations

Students may make basic calculation mistakes when converting percentages to decimals (writing \(0.042\) instead of \(0.42\) for \(42\%\)) or when subtracting the decimal values (\(0.42 - 0.33 = 0.09\)).

2. Incorrectly calculating the market share values

Some students might calculate \(42\%\) of x as \((42/100) \times \mathrm{x}\) instead of \(0.42 \times \mathrm{x}\), or make similar errors with the \(33\%\) calculation, leading to incorrect intermediate results.

Errors while selecting the answer

1. Confusing "\(9\% \text{ of } \mathrm{x}\)" with "\(9\% \text{ of } (\mathrm{x}/100)\)"

After correctly calculating that the decrease is \(0.09\mathrm{x}\), students may not recognize this as "\(9\% \text{ of } \mathrm{x}\)" and instead select one of the "\(9\% \text{ of } (\mathrm{x}/100)\)" options (choice A), which would be 100 times smaller than the correct answer.

2. Selecting \(14\%\) instead of \(9\%\)

Students who made an error earlier in their calculation might arrive at \(14\%\) and select choice E (\(14\% \text{ of } \mathrm{x}\)). This could happen if they incorrectly calculated the market share difference as something other than 9 percentage points.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for x

Since we're dealing with percentages (\(42\%\) and \(33\%\)), let's choose \(\mathrm{x} = 100\) bicycles. This makes percentage calculations straightforward and avoids fractions.

Step 2: Calculate Western European production in 1990

In 1990: Western European producers had \(42\%\) market share
Production in 1990 = \(42\% \text{ of } 100 = 42\) bicycles

Step 3: Calculate Western European production in 1993

In 1993: Western European producers had \(33\%\) market share
Production in 1993 = \(33\% \text{ of } 100 = 33\) bicycles

Step 4: Find the decrease

Decrease = \(42 - 33 = 9\) bicycles

Step 5: Express as a percentage of x

Decrease as percentage of x = \(9/100 = 9\% \text{ of } \mathrm{x}\)

Step 6: Verify with answer choices

Our result is \(9\% \text{ of } \mathrm{x}\), which matches answer choice D.

Why this smart number works:

Choosing \(\mathrm{x} = 100\) is logical because:

• It makes percentage calculations immediate (\(42\% \text{ of } 100 = 42\))
• The relationship between market shares and absolute numbers becomes clear
• The final answer naturally expresses as a clean percentage of x
• Since we're looking for an expression in terms of x, any specific value of x should yield the same proportional relationship