An investor purchased 100 shares of stock X at 61/8 dollars per share and sold them all a year later...

1. Translate the problem requirements: Convert the mixed number stock price to decimal form, understand that brokerage fees reduce both the selling proceeds and increase the purchase cost, and recognize that percent gain compares net profit to net investment
2. Calculate the actual purchase cost including fees: Find the total amount the investor paid out of pocket, including the 2% brokerage fee on the purchase
3. Calculate the actual selling proceeds after fees: Find the net amount the investor received after paying the 2% brokerage fee on the sale
4. Determine the net profit and percent gain: Calculate the difference between proceeds and cost, then find what percent this profit represents of the original investment

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we need to find and organizing our information clearly.

We have an investor who:
• Bought 100 shares at \(6\frac{1}{8}\) dollars per share
• Sold all shares at 24 dollars per share
• Paid 2% brokerage fee on both purchase and sale

First, let's convert that mixed number to a decimal: \(6\frac{1}{8} = 6.125\) dollars per share.

The key insight here is that brokerage fees work against the investor twice:
• When buying: the investor pays MORE than just the stock price
• When selling: the investor receives LESS than just the stock price

We need to find the percent gain, which compares the net profit to the actual amount invested.

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

2. Calculate the actual purchase cost including fees

Let's figure out how much the investor actually paid out of pocket.

Step 1: Calculate the basic purchase price
\(100 \text{ shares} \times \$6.125 \text{ per share} = \$612.50\)

Step 2: Calculate the brokerage fee on purchase
Brokerage fee = \(2\% \text{ of } \$612.50 = 0.02 \times \$612.50 = \$12.25\)

Step 3: Find total amount paid
Total purchase cost = \(\$612.50 + \$12.25 = \$624.75\)

So the investor actually paid $624.75 out of pocket to make this investment.

3. Calculate the actual selling proceeds after fees

Now let's see how much the investor actually received when selling.

Step 1: Calculate the basic selling price
\(100 \text{ shares} \times \$24 \text{ per share} = \$2,400\)

Step 2: Calculate the brokerage fee on sale
Brokerage fee = \(2\% \text{ of } \$2,400 = 0.02 \times \$2,400 = \$48\)

Step 3: Find net proceeds received
Net selling proceeds = \(\$2,400 - \$48 = \$2,352\)

So the investor actually received $2,352 from the sale.

4. Determine the net profit and percent gain

Now we can calculate the actual profit and convert it to a percentage gain.

Step 1: Calculate net profit
Net profit = Net proceeds - Total cost = \(\$2,352 - \$624.75 = \$1,727.25\)

Step 2: Calculate percent gain
Percent gain compares the profit to the original investment:
Percent gain = \(\frac{\text{Net profit}}{\text{Total investment}} \times 100\%\)
Percent gain = \(\frac{\$1,727.25}{\$624.75} \times 100\%\)
Percent gain = \(2.765 \times 100\% = 276.5\%\)

This rounds to approximately 280%.

Process Skill: SIMPLIFY - Breaking down the percent calculation into clear steps

Final Answer

The investor's percent gain is approximately 280%, which matches answer choice (C).

To verify: The investor put in about $625 and made a profit of about $1,727. Since $1,727 is roughly 2.8 times $625, this represents a 280% gain, confirming our answer.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding how brokerage fees affect the investment

Students often forget that brokerage fees reduce the investor's actual returns in two ways: they increase the total purchase cost (you pay more than just the stock price) and decrease the net selling proceeds (you receive less than just the stock price). Some students might only account for fees on one transaction or ignore fees entirely.

2. Confusion about what constitutes the "investment amount" for percent gain calculation

Students may use the basic stock purchase price ($612.50) as their investment base instead of the actual out-of-pocket cost including fees ($624.75). This leads to an incorrect percent gain calculation since the denominator should reflect what the investor truly invested.

Errors while executing the approach

1. Converting mixed number incorrectly

When converting \(6\frac{1}{8}\) to decimal form, students might make errors like writing it as 6.18 instead of 6.125, or incorrectly calculating \(\frac{1}{8}\) as something other than 0.125.

2. Percentage calculation errors with brokerage fees

Students may calculate 2% incorrectly, such as using 2 instead of 0.02 as the multiplier, or they might add the percentage fee to the selling price instead of subtracting it from the proceeds received.

3. Arithmetic mistakes in the final percent gain calculation

The division \(\$1,727.25 \div \$624.75\) requires careful calculation. Students might make computational errors here or forget to multiply by 100 to convert the decimal result to a percentage.

Errors while selecting the answer

1. Selecting an answer without proper rounding consideration

The calculated result of 276.5% needs to be matched to the closest answer choice. Students might select 240% (choice B) or 300% (choice D) without recognizing that 280% (choice C) is actually the closest match to their calculated value.