Alfredo invested a total of $33{,}000 in 2 accounts, A and B, with annual interest rates of 5% and 3%,...

1. Translate the problem requirements: We have \(\$33,000\) split between two accounts with different interest rates (\(5\%\) and \(3\%\)), and we know that Account A earned twice the interest of Account B in the first year. We need to find the total interest earned.
2. Set up variables for the unknown amounts: Define variables for the money invested in each account, keeping the constraint that they sum to \(\$33,000\).
3. Express the interest relationship mathematically: Use the given condition that Account A's interest equals twice Account B's interest to create an equation.
4. Solve for the individual account amounts: Use the constraint equation to find how much was invested in each account.
5. Calculate total interest earned: Find the interest from each account and sum them to get the total.

Execution of Strategic Approach

1. Translate the problem requirements

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

• Alfredo has \(\$33,000\) total to invest
• He splits this money between two accounts: Account A and Account B
• Account A pays \(5\%\) interest per year
• Account B pays \(3\%\) interest per year
• After one year, Account A earned twice as much interest as Account B
• We need to find the total interest earned by both accounts

The key insight is that interest earned = the amount invested × the interest rate. So if Account A earned twice the interest of Account B, it's because of the combination of how much money was in each account AND their different interest rates.

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

2. Set up variables for the unknown amounts

Since we don't know how the \(\$33,000\) was split between the accounts, let's use variables:

• Let \(\mathrm{x}\) = amount invested in Account A
• Since the total is \(\$33,000\), the amount in Account B = \(\$33,000 - \mathrm{x}\)

This is much simpler than using two separate variables because we automatically satisfy the constraint that the amounts must add up to \(\$33,000\).

3. Express the interest relationship mathematically

Now let's calculate the interest earned by each account:

• Interest from Account A = \(\mathrm{x} \times 5\% = 0.05\mathrm{x}\)
• Interest from Account B = \((\$33,000 - \mathrm{x}) \times 3\% = 0.03(\$33,000 - \mathrm{x})\)

The problem tells us that Account A's interest is twice Account B's interest. In plain English: Interest from A = 2 × Interest from B

So our equation is:
\(0.05\mathrm{x} = 2 \times 0.03(\$33,000 - \mathrm{x})\)

4. Solve for the individual account amounts

Let's solve this step by step:

\(0.05\mathrm{x} = 2 \times 0.03(\$33,000 - \mathrm{x})\)
\(0.05\mathrm{x} = 0.06(\$33,000 - \mathrm{x})\)
\(0.05\mathrm{x} = 0.06 \times \$33,000 - 0.06\mathrm{x}\)
\(0.05\mathrm{x} = \$1,980 - 0.06\mathrm{x}\)

Adding \(0.06\mathrm{x}\) to both sides:
\(0.05\mathrm{x} + 0.06\mathrm{x} = \$1,980\)
\(0.11\mathrm{x} = \$1,980\)

Dividing both sides by \(0.11\):
\(\mathrm{x} = \$1,980 ÷ 0.11 = \$18,000\)

So:

• Amount in Account A = \(\$18,000\)
• Amount in Account B = \(\$33,000 - \$18,000 = \$15,000\)

Let's verify this makes sense: Account A has more money AND a higher interest rate, so it should definitely earn more interest than Account B.

5. Calculate total interest earned

Now we can find the interest from each account:

• Interest from Account A = \(\$18,000 \times 5\% = \$18,000 \times 0.05 = \$900\)
• Interest from Account B = \(\$15,000 \times 3\% = \$15,000 \times 0.03 = \$450\)

Let's check: Is Account A's interest twice Account B's interest?
\(\$900 = 2 \times \$450 = \$900\) ✓

Total interest earned = \(\$900 + \$450 = \$1,350\)

Final Answer

The total interest earned by both accounts for the first year is \(\$1,350\).

Looking at our answer choices, this matches choice C exactly.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the "twice the interest" relationship: Students often confuse what "twice" refers to. They might think Account A has twice the money as Account B, or twice the interest rate, rather than understanding that Account A earned twice the interest amount. This leads to setting up incorrect equations like \(\mathrm{x} = 2(33,000 - \mathrm{x})\) instead of \(0.05\mathrm{x} = 2 \times 0.03(33,000 - \mathrm{x})\).

2. Using two separate variables unnecessarily: Some students define \(\mathrm{x}\) = amount in Account A and \(\mathrm{y}\) = amount in Account B, then forget to use the constraint \(\mathrm{x} + \mathrm{y} = 33,000\). This creates an system that's harder to solve and increases chances of algebraic errors. The cleaner approach is to use just one variable and express the second amount as \((33,000 - \mathrm{x})\).

3. Confusing simple interest vs. compound interest: While this problem uses simple interest (which is straightforward), students sometimes overthink and try to apply compound interest formulas, making the problem unnecessarily complex.

Errors while executing the approach

1. Arithmetic errors in decimal multiplication: When calculating \(0.06 \times 33,000\), students often make mistakes with decimal placement, getting \(198\) instead of \(1,980\), or when dividing \(1,980\) by \(0.11\), incorrectly getting \(1,800\) instead of \(18,000\).

2. Sign errors when moving terms across the equation: In the step \(0.05\mathrm{x} = 1,980 - 0.06\mathrm{x}\), students sometimes incorrectly write \(0.05\mathrm{x} - 0.06\mathrm{x} = 1,980\) instead of \(0.05\mathrm{x} + 0.06\mathrm{x} = 1,980\), forgetting that moving \(-0.06\mathrm{x}\) to the left side changes it to \(+0.06\mathrm{x}\).

3. Forgetting to verify the solution: Students solve for \(\mathrm{x} = 18,000\) but don't check whether Account A's interest \((900)\) is actually twice Account B's interest \((450)\). This verification step catches calculation errors and confirms the logic is correct.

Errors while selecting the answer

1. Stopping at individual account interest instead of total: After calculating that Account A earns \(\$900\) and Account B earns \(\$450\), some students select \(\$900\) as their answer because it's the larger amount, forgetting that the question asks for the total interest earned by both accounts combined.