Carl invested a certain amount of money for one year, some of which earned interest at a simple annual interest...

1. Translate the problem requirements: Carl splits his total investment between two portions - one earning 2.2% simple interest and another earning 2.5% simple interest. Together, these portions generated $680 in total interest for one year. We need to determine which of the given amounts could be his total investment.
2. Set up the mathematical relationship: Express the total interest as a function of how the money is split between the two interest rates, using the fact that if x dollars earn 2.2% and (Total - x) dollars earn 2.5%, then total interest equals \(0.022\mathrm{x} + 0.025(\mathrm{Total} - \mathrm{x})\).
3. Establish the constraints: Since the interest earned ($680) must fall between the minimum possible (if all money earned 2.2%) and maximum possible (if all money earned 2.5%), determine the range of valid total investments.
4. Test each given option: For each proposed total investment amount, check if it's possible to split the money between the two rates to achieve exactly $680 in interest, ensuring both portions are non-negative.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what's happening here in plain English. Carl has a pot of money that he wants to invest for one year. He decides to split this money into two portions:

• One portion earns 2.2% simple interest per year
• The remaining portion earns 2.5% simple interest per year

At the end of the year, the total interest from both portions combined is $680.

We need to figure out which of the three given amounts ($24,708, $27,896, or $31,284) could have been Carl's total investment amount.

Let's call Carl's total investment "T" dollars. If he puts "x" dollars at 2.2%, then he puts "(T - x)" dollars at 2.5%.

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

2. Set up the mathematical relationship

Now let's think about how interest works. With simple interest:

• Interest from the 2.2% portion = \(\mathrm{x} \times 0.022\)
• Interest from the 2.5% portion = \((\mathrm{T} - \mathrm{x}) \times 0.025\)

The total interest equation becomes:

\(0.022\mathrm{x} + 0.025(\mathrm{T} - \mathrm{x}) = 680\)

Let's expand this: \(0.022\mathrm{x} + 0.025\mathrm{T} - 0.025\mathrm{x} = 680\)

Combining like terms: \(0.025\mathrm{T} - 0.003\mathrm{x} = 680\)

Solving for x: \(\mathrm{x} = (0.025\mathrm{T} - 680) \div 0.003\)

3. Establish the constraints

Here's the key insight: Carl can't invest negative amounts in either portion. This means:

• \(\mathrm{x} \geq 0\) (amount at 2.2% must be non-negative)
• \((\mathrm{T} - \mathrm{x}) \geq 0\), which means \(\mathrm{x} \leq \mathrm{T}\) (amount at 2.5% must be non-negative)

Let's think about this practically. What are the extreme cases?

Minimum interest scenario: If all money earned 2.2%, total interest = \(\mathrm{T} \times 0.022\)

Maximum interest scenario: If all money earned 2.5%, total interest = \(\mathrm{T} \times 0.025\)

Since Carl earned $680, we need: \(\mathrm{T} \times 0.022 \leq 680 \leq \mathrm{T} \times 0.025\)

This gives us: \(680 \div 0.025 \leq \mathrm{T} \leq 680 \div 0.022\)

Calculating: \(\$27,200 \leq \mathrm{T} \leq \$30,909\) (approximately)

Process Skill: APPLY CONSTRAINTS - Using logical boundaries to limit possible solutions

4. Test each given option

Now let's check each option against our constraints and see if we can actually split the money to get $680 interest.

Option I: T = $24,708

This is less than our minimum of $27,200, so it's impossible. Even if all money earned the higher rate of 2.5%, total interest would only be: \(\$24,708 \times 0.025 = \$617.70\), which is less than $680.

Option II: T = $27,896

This falls within our range ($27,200 to $30,909). Let's check if we can split it:

Using \(\mathrm{x} = (0.025\mathrm{T} - 680) \div 0.003\):

\(\mathrm{x} = (0.025 \times 27,896 - 680) \div 0.003 = (697.40 - 680) \div 0.003 = 17.40 \div 0.003 = \$5,800\)

So Carl would put $5,800 at 2.2% and $22,096 at 2.5%.

Check: \(\$5,800 \times 0.022 + \$22,096 \times 0.025 = \$127.60 + \$552.40 = \$680\) ✓

Both amounts are positive, so this works!

Option III: T = $31,284

This exceeds our maximum of $30,909, so let's check:

\(\mathrm{x} = (0.025 \times 31,284 - 680) \div 0.003 = (782.10 - 680) \div 0.003 = 102.10 \div 0.003 = \$34,033\)

This would require putting $34,033 at 2.2%, but Carl only has $31,284 total. This is impossible!

4. Final Answer

Only Option II ($27,896) works. Carl could invest $5,800 at 2.2% and $22,096 at 2.5% to earn exactly $680 in interest.

The answer is B. II only

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint boundaries: Students often fail to recognize that the total interest earned ($680) must fall between the minimum possible interest (if all money earned 2.2%) and maximum possible interest (if all money earned 2.5%). This leads them to attempt calculations on investment amounts that are mathematically impossible, like trying to verify $24,708 without first checking if it could even generate $680 in interest.

2. Setting up incorrect variable relationships: Students frequently struggle with defining the two portions of investment correctly. They might set up the equation as x + y = T and then separately define interest equations, instead of recognizing that if x dollars go to 2.2%, then automatically (T - x) dollars must go to 2.5%. This leads to overcomplicated systems with unnecessary variables.

3. Overlooking non-negativity constraints: Many students forget that both investment portions must be positive amounts (x ≥ 0 and T - x ≥ 0). They set up the interest equation correctly but fail to establish that their solution must result in realistic, positive amounts for both investment portions.

Errors while executing the approach

1. Arithmetic errors in boundary calculations: When calculating the minimum and maximum possible investment amounts, students often make errors with division: incorrectly computing \(680 \div 0.022 \approx 30,909\) or \(680 \div 0.025 = 27,200\). These boundary errors lead them to incorrectly eliminate valid options or include impossible ones.

2. Algebraic manipulation mistakes: Students frequently make errors when solving for x in the equation \(\mathrm{x} = (0.025\mathrm{T} - 680) \div 0.003\). Common mistakes include incorrect order of operations, wrong signs when expanding \(0.025(\mathrm{T} - \mathrm{x})\), or errors when isolating the variable x.

3. Verification calculation errors: Even when students arrive at the correct split amounts, they often make arithmetic errors during verification. For example, calculating \(\$5,800 \times 0.022\) or \(\$22,096 \times 0.025\) incorrectly, leading them to think a valid solution doesn't work or that an invalid solution is correct.

Errors while selecting the answer

1. Misreading the Roman numeral format: Students correctly identify that only option II works but then confuse the answer choices. They might select 'D. I and II' thinking that since II works and they didn't carefully check I, or select 'C. III only' by mixing up which Roman numeral corresponds to which dollar amount.

2. Incomplete option testing: Students sometimes stop testing after finding one option that works (option II) and assume other options might also work without fully verifying them. This leads to selecting 'E. II and III' or 'D. I and II' instead of the correct 'B. II only' because they didn't complete their analysis.