The closing price of Stock X changed on each trading day last month. The percent change in the closing price...

1. Translate the problem requirements: The stock price changed every day, and from the first day to any other day, the percent change was less than 50%. We know the second day price was $10, and we need to find which final day price is impossible.
2. Establish the constraint boundaries: Determine what "less than 50% change from first day" means in terms of actual price limits for the first day, given that day 2 was $10.
3. Find the possible range for the first day price: Use the constraint that the change from first day to second day must be less than 50% to establish minimum and maximum possible values for the first day price.
4. Determine the possible range for the last day price: Apply the same constraint (less than 50% change from first day) to find what range of final day prices is theoretically possible.
5. Test each answer choice: Check which answer choice falls outside the possible range we established.

Execution of Strategic Approach

1. Translate the problem requirements

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

• The stock price changed every single trading day last month
• From the first trading day to ANY other trading day, the percent change was less than 50%
• On the second trading day, the price was $10.00
• We need to find which final day price is impossible

The key insight here is understanding what "less than 50% change" means. If a price goes up by less than 50%, it means it's less than 1.5 times the original. If it goes down by less than 50%, it means it's more than 0.5 times the original.

Process Skill: TRANSLATE - Converting the constraint language into concrete mathematical boundaries

2. Establish the constraint boundaries

Let's think about this step by step. If the first day price was \(\mathrm{P_1}\), then:

• A price increase of less than 50% means: New Price < \(\mathrm{P_1} \times 1.5\)
• A price decrease of less than 50% means: New Price > \(\mathrm{P_1} \times 0.5\)

So for ANY day after the first day, the price must be between \(\mathrm{P_1} \times 0.5\) and \(\mathrm{P_1} \times 1.5\) (not including these exact values).

3. Find the possible range for the first day price

We know the second day price was $10.00, and this must satisfy our constraint:
\(\mathrm{P_1} \times 0.5 < \$10.00 < \mathrm{P_1} \times 1.5\)

Let's solve this:

• From \(\mathrm{P_1} \times 0.5 < \$10.00\): \(\mathrm{P_1} < \$20.00\)
• From \(\$10.00 < \mathrm{P_1} \times 1.5\): \(\mathrm{P_1} > \$6.67\) (approximately)

So the first day price must be between $6.67 and $20.00 (approximately).

Process Skill: INFER - Drawing the conclusion about possible first day prices from the constraint

4. Determine the possible range for the last day price

Now, the last day price must also satisfy the constraint relative to the first day price:
\(\mathrm{P_1} \times 0.5 < \text{Last Day Price} < \mathrm{P_1} \times 1.5\)

Since \(\mathrm{P_1}\) can range from about $6.67 to $20.00:

• Minimum possible last day price: \(\$6.67 \times 0.5 = \$3.33\) (approximately)
• Maximum possible last day price: \(\$20.00 \times 1.5 = \$30.00\)

But remember, these are strict inequalities (less than 50%, not equal to), so the actual range is slightly smaller than $3.33 to $30.00.

5. Test each answer choice

Let's check each option against our possible range:

• A. $3.00 - This is LESS than our minimum of approximately $3.33, so this is impossible
• B. $9.00 - This falls within our possible range
• C. $19.00 - This falls within our possible range
• D. $24.00 - This falls within our possible range
• E. $29.00 - This falls within our possible range

Process Skill: APPLY CONSTRAINTS - Systematically checking each option against our established boundaries

Final Answer

Choice A ($3.00) CANNOT be the closing price on the last trading day because it falls outside the possible range we established. All other choices are theoretically possible given the constraints.

The answer is A.

Common Faltering Points

Errors while devising the approach

• Misinterpreting the constraint scope: Students often miss that the "less than 50% change" constraint applies from the FIRST day to EVERY other day (including the last day), not just between consecutive days. They might think it only applies between the first and second day, leading to an incorrect approach that doesn't establish the proper boundaries for the last day's price.
• Confusing "less than 50%" with "at most 50%": Students may treat "less than 50%" as "≤ 50%" instead of "< 50%", which would incorrectly include the boundary values (\(0.5\mathrm{P_1}\) and \(1.5\mathrm{P_1}\)) as possible prices. This subtle difference affects the final range calculations.
• Working backwards incorrectly: Students might attempt to work backwards from each answer choice to see if it's possible, rather than establishing the theoretical range first. This approach often leads to incomplete analysis since they may not properly connect the constraints between all three days (first, second, and last).

Errors while executing the approach

• Inequality direction errors: When solving \(\mathrm{P_1} \times 0.5 < \$10.00 < \mathrm{P_1} \times 1.5\), students frequently make errors in flipping inequality signs or solving for \(\mathrm{P_1}\). For example, from \(\mathrm{P_1} \times 0.5 < \$10.00\), they might incorrectly conclude \(\mathrm{P_1} > \$20.00\) instead of \(\mathrm{P_1} < \$20.00\).
• Boundary calculation mistakes: Students often make arithmetic errors when calculating the extreme values, such as computing \(\$20.00 \div 3\) instead of \(\$20.00 \div 1.5\) for the minimum first day price, or incorrectly calculating the minimum possible last day price as \(\$6.67 \times 0.5\).
• Range endpoint confusion: Students may correctly find the range for \(\mathrm{P_1}\) but then incorrectly apply the constraints to find the last day range, forgetting that they need to use the extreme values of \(\mathrm{P_1}\) to establish the extreme possible values for the last day.

Errors while selecting the answer

• Choosing a possible value instead of impossible: Students may correctly identify that $3.00 is very close to the boundary but mistakenly conclude it's still possible, especially if they made small errors in their boundary calculations. They might select an answer choice that clearly falls within the acceptable range.
• Misreading "CANNOT be": Students sometimes miss that the question asks for what CANNOT be the price and instead select an answer that CAN be the price. This is a classic error where students solve the problem correctly but select the wrong type of answer.