Victor invested a total of $800 in 2 stocks. At the end of 1 year, one stock had gained 12%...

1. Translate the problem requirements: Victor splits \(\$800\) between two stocks. After one year, one stock gains \(12\%\) and the other loses \(4\%\). The combined value becomes \(\$848\). We need to find how much was invested in the stock that lost value.
2. Set up variables for the two investments: Define variables for the amount invested in each stock, ensuring they sum to \(\$800\).
3. Express the final values after percentage changes: Calculate what each investment becomes after the \(12\%\) gain and \(4\%\) loss, then set up an equation where their sum equals \(\$848\).
4. Solve for the investment amounts: Use algebraic manipulation to find the specific amount invested in the stock that decreased in value.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in everyday terms: Victor starts with \(\$800\) total to invest. He splits this money between two different stocks. After one year passes, one stock performs well (gaining \(12\%\) in value) while the other performs poorly (losing \(4\%\) in value). When Victor checks his portfolio, the combined value of both stocks is now \(\$848\).

Our goal is to find out how much money Victor originally put into the stock that lost value (the one that decreased by \(4\%\)).

Think of it like this: if Victor put more money in the winning stock, his total would be higher. If he put more in the losing stock, his total would be lower. Since his portfolio grew from \(\$800\) to \(\$848\), we need to find the exact split that produces this result.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up variables for the two investments

Let's use simple language to define what we're looking for:

- Let's call the amount invested in the stock that LOST \(4\%\) as 'x' dollars
- Since Victor invested a total of \(\$800\), the amount in the other stock (the one that GAINED \(12\%\)) must be \((800 - \mathrm{x})\) dollars

This makes sense because: amount in losing stock + amount in winning stock = total investment
So: \(\mathrm{x} + (800 - \mathrm{x}) = 800\) ✓

In mathematical notation:

- Investment in stock that lost \(4\%\) = \(\mathrm{x}\)
- Investment in stock that gained \(12\%\) = \((800 - \mathrm{x})\)

3. Express the final values after percentage changes

Now let's figure out what each investment is worth after one year:

For the stock that lost \(4\%\):

- Original investment: \(\mathrm{x}\) dollars
- After losing \(4\%\), it's worth \(96\%\) of original value
- Final value = \(\mathrm{x} \times 0.96\)

For the stock that gained \(12\%\):

- Original investment: \((800 - \mathrm{x})\) dollars
- After gaining \(12\%\), it's worth \(112\%\) of original value
- Final value = \((800 - \mathrm{x}) \times 1.12\)

Since the total portfolio value is \(\$848\):
Value of losing stock + Value of winning stock = \(\$848\)

Therefore: \(\mathrm{x} \times 0.96 + (800 - \mathrm{x}) \times 1.12 = 848\)

4. Solve for the investment amounts

Let's solve this equation step by step:

\(\mathrm{x} \times 0.96 + (800 - \mathrm{x}) \times 1.12 = 848\)

First, let's expand the second term:
\(0.96\mathrm{x} + 800 \times 1.12 - \mathrm{x} \times 1.12 = 848\)
\(0.96\mathrm{x} + 896 - 1.12\mathrm{x} = 848\)

Combine like terms:
\((0.96 - 1.12)\mathrm{x} + 896 = 848\)
\(-0.16\mathrm{x} + 896 = 848\)

Subtract 896 from both sides:
\(-0.16\mathrm{x} = 848 - 896\)
\(-0.16\mathrm{x} = -48\)

Divide both sides by \(-0.16\):
\(\mathrm{x} = -48 \div (-0.16)\)
\(\mathrm{x} = 300\)

Let's verify: If Victor invested \(\$300\) in the losing stock and \(\$500\) in the winning stock:

- Losing stock final value: \(\$300 \times 0.96 = \$288\)
- Winning stock final value: \(\$500 \times 1.12 = \$560\)
- Total final value: \(\$288 + \$560 = \$848\) ✓

Process Skill: MANIPULATE - Algebraic manipulation to solve for the unknown

4. Final Answer

Victor invested \(\$300\) in the stock that decreased in value.

This matches answer choice C: \(\$300\).

Common Faltering Points

Errors while devising the approach

1. Misidentifying which stock to find

Students often confuse what the question is asking for. The question asks "How much did Victor invest in the stock that decreased in value?" but students may set up their variable x to represent the stock that gained \(12\%\) instead of the one that lost \(4\%\). This leads to finding the wrong investment amount and selecting an incorrect answer.

2. Incorrect setup of the constraint equation

Some students struggle with the total investment constraint. They might set up the equation as \(\mathrm{x} + \mathrm{y} = 800\) and then try to solve for two unknowns instead of recognizing that if one investment is x, the other must be \((800 - \mathrm{x})\). This approach makes the problem unnecessarily complex and often leads to getting stuck.

3. Misunderstanding percentage changes

Students frequently confuse how to express percentage gains and losses mathematically. They might represent a \(4\%\) loss as multiplying by \(0.04\) instead of \(0.96\), or a \(12\%\) gain as multiplying by \(0.12\) instead of \(1.12\). This fundamental error in translating percentages carries through the entire solution.

Errors while executing the approach

1. Sign errors when combining like terms

When simplifying the equation \(0.96\mathrm{x} + 896 - 1.12\mathrm{x} = 848\), students often make sign errors. They might incorrectly combine the x terms as \((0.96 + 1.12)\mathrm{x}\) instead of \((0.96 - 1.12)\mathrm{x}\), leading to \(+0.08\mathrm{x}\) instead of \(-0.16\mathrm{x}\), which completely changes the solution.

2. Arithmetic mistakes in decimal calculations

The problem involves several decimal calculations (\(0.96, 1.12, -0.16\)). Students commonly make arithmetic errors such as calculating \(800 \times 1.12\) incorrectly or making mistakes when dividing \(-48\) by \(-0.16\). These calculation errors lead to wrong final answers even with correct setup.

3. Forgetting to distribute correctly

When expanding \((800 - \mathrm{x}) \times 1.12\), students sometimes forget to distribute the \(1.12\) to both terms, writing it as \(800 \times 1.12 - \mathrm{x}\) instead of \(800 \times 1.12 - \mathrm{x} \times 1.12\). This distribution error significantly affects the equation and final answer.

Errors while selecting the answer

1. Selecting the wrong investment amount

After correctly solving for \(\mathrm{x} = 300\), some students mistakenly select the answer that represents the other stock's investment (\(\$500\)) instead of the amount invested in the stock that decreased in value (\(\$300\)). This happens when they lose track of what their variable x represents.

2. Not verifying the answer against the original problem

Students might arrive at a numerical answer but fail to check whether it makes logical sense. For instance, if their calculation gives a negative value or an amount greater than \(\$800\), they should recognize this as impossible, but often proceed to select an answer choice anyway without this sanity check.