Bob invested $2000 in fund A and $1000 in fund B. Over the next two years, the money in Fund...

1. Translate the problem requirements: We need to find the final value of each investment after 2 years, then calculate how much more Fund A is worth than Fund B. Fund A earns 12% total over 2 years (simple interest), while Fund B earns 30% annual interest compounded annually.
2. Calculate Fund A's final value: Apply the 12% total interest to the \(\$2000\) initial investment over the 2-year period.
3. Calculate Fund B's final value: Apply 30% annual compound interest to the \(\$1000\) initial investment for 2 years, meaning we calculate \(1.30 \times 1.30 = 1.69\) times the original amount.
4. Find the difference: Subtract Fund B's final value from Fund A's final value to determine how much more Fund A is worth.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to find in everyday language:

• Bob puts \(\$2000\) in Fund A and \(\$1000\) in Fund B
• Fund A grows by 12% total over the entire 2-year period (this is simple interest)
• Fund B grows by 30% each year, and the growth compounds (meaning the second year's 30% applies to the already-grown amount from year 1)
• We need to find how much MORE Fund A is worth than Fund B after 2 years

Think of it like this: if you have two piggy banks that grow differently, we want to know the difference between what's in each piggy bank after 2 years.

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

2. Calculate Fund A's final value

Fund A starts with \(\$2000\) and earns 12% total interest over 2 years.

In plain English: if something grows by 12%, it becomes 112% of its original value (100% + 12% = 112%).

So Fund A becomes: \(\$2000 \times 1.12 = \$2240\)

This means Fund A gained \(\$240\) in interest (\(\$2240 - \$2000 = \$240\)).

3. Calculate Fund B's final value

Fund B starts with \(\$1000\) and grows by 30% each year with compounding.

Let's think through this year by year:

• After Year 1: \(\$1000\) grows by 30%, so it becomes \(\$1000 \times 1.30 = \$1300\)
• After Year 2: The \(\$1300\) grows by another 30%, so it becomes \(\$1300 \times 1.30 = \$1690\)

We can also calculate this directly: \(\$1000 \times 1.30 \times 1.30 = \$1000 \times 1.69 = \$1690\)

This means Fund B gained \(\$690\) in interest (\(\$1690 - \$1000 = \$690\)).

4. Find the difference

Now we compare the final values:

• Fund A final value: \(\$2240\)
• Fund B final value: \(\$1690\)
• Difference: \(\$2240 - \$1690 = \$550\)

Therefore, Bob's investment in Fund A is worth \(\$550\) more than his investment in Fund B.

Final Answer

The answer is B. \(\$550\)

We can verify this makes sense: even though Fund B had a higher growth rate (30% annually vs 12% total), Fund A started with twice as much money (\(\$2000\) vs \(\$1000\)), so it ends up with a higher final value.

Common Faltering Points

Errors while devising the approach

Students often assume that "12 percent for the two years combined" means 12% per year, leading them to calculate Fund A's value as \(\$2000 \times 1.12 \times 1.12 = \$2508.80\) instead of the correct \(\$2000 \times 1.12 = \$2240\). The phrase "for the two years combined" clearly indicates this is total interest over both years, not annual interest.

Students may misread Fund A as having compound interest like Fund B, especially since Fund B explicitly mentions "compounded annually" while Fund A doesn't specify the interest type. However, when a problem states total percentage over a period without mentioning compounding, it typically means simple interest applied once.

Some students might set up to find the total value of each fund or the percentage difference, rather than the absolute dollar difference between Fund A and Fund B. The question specifically asks "how much more" Fund A is worth than Fund B, which requires subtracting Fund B's final value from Fund A's final value.

Errors while executing the approach

When calculating Fund B's value, students often make mistakes with repeated multiplication: \(\$1000 \times 1.30 \times 1.30\). Common errors include calculating \(1.30^2\) as 1.60 instead of 1.69, or making basic multiplication errors like \(\$1000 \times 1.69 = \$1600\) instead of \(\$1690\).

Students frequently convert 30% to 0.30 but then add it incorrectly, using \(\$1000 \times (1 + 0.30)\) but calculating it as \(\$1000 \times 1.03\) instead of \(\$1000 \times 1.30\), or miscalculating the decimal equivalent of percentages under time pressure.

Errors while selecting the answer

After correctly calculating Fund A = \(\$2240\) and Fund B = \(\$1690\), students might mistakenly select \(\$1690\) as their answer instead of performing the final subtraction to get \(\$550\). This happens because they lose track of what the question is asking after focusing intensely on the calculations.

Students might calculate Fund B - Fund A = \(\$1690 - \$2240 = -\$550\) and then select \(\$550\), or they might get confused about which fund should be larger and select an answer that assumes Fund B is worth more than Fund A due to its higher interest rate.