Last year, sales at Company X were 10% greater in February than in January, 15% less in March than in...

1. Translate the problem requirements: We need to track sales through 6 months starting from January as our baseline. Each month has a percentage change from the previous month: Feb (+\(10\%\)), Mar (-\(15\%\)), Apr (+\(20\%\)), May (-\(10\%\)), June (+\(5\%\)). We need to find which month's sales value is closest to January's original value.
2. Establish a baseline and track cumulative changes: Set January sales as our reference point (100 or any convenient number) and calculate the actual sales value for each subsequent month by applying the percentage changes sequentially.
3. Calculate month-by-month sales values: Apply each percentage change to the previous month's result to get the running sales figures for Feb through June.
4. Compare distances from the January baseline: Calculate how far each month's sales are from January's sales and identify the month with the smallest difference.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're actually looking for here. We have Company X's sales data over 6 months, and we need to track how the sales change from month to month.

Think of it like this: January is our starting point - whatever the sales were that month, we'll call that our "home base." Then each month after that, sales go up or down by a certain percentage compared to the previous month.

Here's what happens each month:

• February: Sales go UP by \(10\%\) from January
• March: Sales go DOWN by \(15\%\) from February
• April: Sales go UP by \(20\%\) from March
• May: Sales go DOWN by \(10\%\) from April
• June: Sales go UP by \(5\%\) from May

Our job is to figure out which of these months (Feb through June) ends up with sales closest to where we started in January.

Process Skill: TRANSLATE - Converting the percentage change language into a clear mathematical tracking problem

2. Establish a baseline and track cumulative changes

To make the math simple, let's say January sales were exactly 100 units. This makes it easy to see what's happening with percentages.

So our baseline is:

• January = 100 units

Now we need to understand that each month's change builds on the previous month. It's like a chain reaction - each change affects the next calculation.

For example:

• If February is \(10\%\) higher than January's 100, then February = 110
• If March is \(15\%\) lower than February's 110, then March = 110 minus \(15\%\) of 110

This step-by-step approach ensures we don't miss any of the cumulative effects.

3. Calculate month-by-month sales values

Now let's work through each month carefully:

January: 100 (our baseline)

February: \(10\%\) greater than January
\(10\% \times 100 = 10\)
\(\mathrm{February} = 100 + 10 = 110\)

March: \(15\%\) less than February
\(15\% \times 110 = 16.5\)
\(\mathrm{March} = 110 - 16.5 = 93.5\)

April: \(20\%\) greater than March
\(20\% \times 93.5 = 18.7\)
\(\mathrm{April} = 93.5 + 18.7 = 112.2\)

May: \(10\%\) less than April
\(10\% \times 112.2 = 11.22\)
\(\mathrm{May} = 112.2 - 11.22 = 100.98\)

June: \(5\%\) greater than May
\(5\% \times 100.98 = 5.049\)
\(\mathrm{June} = 100.98 + 5.049 = 106.029\)

Process Skill: SIMPLIFY - Using 100 as baseline makes percentage calculations straightforward

4. Compare distances from the January baseline

Now we need to see which month's sales are closest to January's sales of 100.

Let's calculate how far each month is from our January baseline of 100:

• February: \(|110 - 100| = 10\) units away
• March: \(|93.5 - 100| = 6.5\) units away
• April: \(|112.2 - 100| = 12.2\) units away
• May: \(|100.98 - 100| = 0.98\) units away
• June: \(|106.029 - 100| = 6.029\) units away

Looking at these differences:

• May is only 0.98 units away from January
• March is 6.5 units away
• June is about 6 units away
• February is 10 units away
• April is 12.2 units away

Clearly, May at 100.98 is extremely close to January's 100 - much closer than any other month.

5. Final Answer

May has sales of 100.98, which is closest to January's baseline of 100, with a difference of less than 1 unit.

The answer is D. May

Common Faltering Points

Errors while devising the approach

1. Misinterpreting percentage base references
Students often misread the problem and think all percentage changes are relative to January, rather than understanding that each month's change is relative to the previous month. For example, they might calculate March as "\(15\%\) less than January" instead of "\(15\%\) less than February." This fundamental misunderstanding leads to completely incorrect calculations throughout the problem.

2. Confusion about what "closest to January" means
Some students may think they need to find which month has the smallest percentage change from the previous month, rather than understanding they need to compare absolute sales values to January's baseline. This misinterpretation would lead them to focus on individual monthly changes rather than cumulative effects.

Errors while executing the approach

1. Arithmetic errors in percentage calculations
Students frequently make calculation mistakes when computing percentages of non-round numbers. For example, when calculating "\(15\% \times 110\)" or "\(20\% \times 93.5\)," they may rush through the arithmetic and get incorrect intermediate values, which then compound through subsequent calculations.

2. Forgetting to track cumulative effects properly
Even when students understand the approach, they may lose track of which value to use as the base for the next calculation. For instance, after correctly calculating March = 93.5, they might mistakenly use January's value (100) instead of March's value (93.5) when calculating April's \(20\%\) increase.

3. Rounding errors affecting final comparison
Students may round intermediate calculations too aggressively (e.g., rounding May to exactly 101 instead of 100.98), which can affect the final comparison of distances from January and potentially lead to selecting the wrong answer.

Errors while selecting the answer

1. Comparing wrong values for "closest"
After calculating all monthly values correctly, students might compare the raw sales numbers instead of the distances from January. For example, they might think March (93.5) is closest to January because it's the smallest number, rather than recognizing that May (100.98) has the smallest absolute difference from 100.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient baseline value
Let's set January sales = 100 units. This round number makes percentage calculations straightforward and easy to track.

Step 2: Calculate sales for each month using the percentage changes

February: \(10\%\) greater than January
\(\mathrm{Feb} = 100 + (10\% \times 100) = 100 + 10 = 110\)

March: \(15\%\) less than February
\(\mathrm{Mar} = 110 - (15\% \times 110) = 110 - 16.5 = 93.5\)

April: \(20\%\) greater than March
\(\mathrm{Apr} = 93.5 + (20\% \times 93.5) = 93.5 + 18.7 = 112.2\)

May: \(10\%\) less than April
\(\mathrm{May} = 112.2 - (10\% \times 112.2) = 112.2 - 11.22 = 100.98\)

June: \(5\%\) greater than May
\(\mathrm{June} = 100.98 + (5\% \times 100.98) = 100.98 + 5.049 = 106.029\)

Step 3: Calculate the distance from January (100) for each month

• February: \(|110 - 100| = 10\)
• March: \(|93.5 - 100| = 6.5\)
• April: \(|112.2 - 100| = 12.2\)
• May: \(|100.98 - 100| = 0.98\)
• June: \(|106.029 - 100| = 6.029\)

Step 4: Identify the smallest distance
May has the smallest distance from January's sales (0.98), making it the closest month to January's sales level.

Answer: D (May)