Janice made three investments totaling $5,000. In the first year the investments earned 6%, 8%, and 9% interest, respectively. Which...

1. Translate the problem requirements: We have \(\$5{,}000\) total invested across three investments earning \(6\%\), \(8\%\), and \(9\%\) respectively. We need to determine which of the given interest amounts (\(\$475\), \(\$425\), \(\$335\)) are possible based on different ways to allocate the \(\$5{,}000\).
2. Establish the theoretical boundaries: Calculate the minimum and maximum possible total interest by considering extreme allocation scenarios (all money in lowest rate vs highest rate).
3. Test each given amount for feasibility: For each proposed interest amount, determine if there exists a valid allocation of the \(\$5{,}000\) that would produce exactly that amount of interest.
4. Eliminate impossible scenarios: Use the boundaries and algebraic constraints to identify which interest amounts cannot be achieved with any valid allocation.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find.

We have \(\$5{,}000\) total that Janice split among three different investments. Think of this like putting money into three different savings accounts:

• Investment 1 earns \(6\%\) interest per year
• Investment 2 earns \(8\%\) interest per year
• Investment 3 earns \(9\%\) interest per year

The key insight is that Janice can put ANY amount into each investment, as long as the three amounts add up to exactly \(\$5{,}000\). She could put \(\$1{,}000\) in each, or \(\$4{,}000\) in one and \(\$500\) in each of the others, or any other combination.

We need to determine which of these total interest amounts are possible: \(\$475\), \(\$425\), or \(\$335\).

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

2. Establish the theoretical boundaries

To understand what's possible, let's think about the extreme cases - what's the most and least interest Janice could earn?

Maximum possible interest: If Janice puts ALL \(\$5{,}000\) into the highest-earning investment (\(9\%\)), she would earn: \(\$5{,}000 \times 0.09 = \$450\)

Minimum possible interest: If Janice puts ALL \(\$5{,}000\) into the lowest-earning investment (\(6\%\)), she would earn: \(\$5{,}000 \times 0.06 = \$300\)

This tells us that ANY total interest between \(\$300\) and \(\$450\) should be possible by mixing the investments appropriately.

Looking at our options:

• \(\$475\) is ABOVE \(\$450\), so this seems impossible
• \(\$425\) is between \(\$300\) and \(\$450\), so this should be possible
• \(\$335\) is between \(\$300\) and \(\$450\), so this should be possible

3. Test each given amount for feasibility

Now let's verify our reasoning by checking if we can actually achieve each amount.

Testing \(\$475\): Since our maximum possible is only \(\$450\), earning \(\$475\) is impossible. We can eliminate option I.

Testing \(\$425\): Let's see if we can find amounts that work. If we let x, y, and z be the amounts invested at \(6\%\), \(8\%\), and \(9\%\) respectively:

• \(x + y + z = 5{,}000\) (total investment)
• \(0.06x + 0.08y + 0.09z = 425\) (total interest)

One way to check: let's try putting most money in the \(9\%\) investment. If we put \(\$4{,}000\) at \(9\%\) and \(\$1{,}000\) at \(8\%\):

Interest = \(\$4{,}000 \times 0.09 + \$1{,}000 \times 0.08 = \$360 + \$80 = \$440\)

If we put \(\$3{,}000\) at \(9\%\), \(\$2{,}000\) at \(8\%\):

Interest = \(\$3{,}000 \times 0.09 + \$2{,}000 \times 0.08 = \$270 + \$160 = \$430\)

Since we got \(\$430\) and \(\$440\), and these are close to \(\$425\), we can fine-tune to get exactly \(\$425\). Option II is possible.

Testing \(\$335\): Let's try putting most money in the lower-earning investments. If we put \(\$3{,}000\) at \(6\%\) and \(\$2{,}000\) at \(8\%\):

Interest = \(\$3{,}000 \times 0.06 + \$2{,}000 \times 0.08 = \$180 + \$160 = \$340\)

This is close to \(\$335\), so with some adjustment we can achieve exactly \(\$335\). Option III is possible.

4. Eliminate impossible scenarios

Based on our boundary analysis and testing:

• Option I (\(\$475\)): IMPOSSIBLE - exceeds the theoretical maximum of \(\$450\)
• Option II (\(\$425\)): POSSIBLE - within our range and we showed feasible allocations
• Option III (\(\$335\)): POSSIBLE - within our range and we showed feasible allocations

Process Skill: APPLY CONSTRAINTS - Using the mathematical boundaries to eliminate impossible answers

5. Final Answer

Only options II and III are possible (II and III only).

The correct answer is D) II and III only.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the investment constraint: Students may think that Janice must put equal amounts (\(\$1{,}667\) each) into all three investments, rather than understanding she can allocate ANY amounts as long as they total \(\$5{,}000\). This rigid thinking prevents them from exploring the full range of possible interest earnings.

2. Failing to establish theoretical boundaries: Students often jump directly to testing specific values without first determining the minimum and maximum possible interest earnings. Without calculating that interest must fall between \(\$300\) (all money at \(6\%\)) and \(\$450\) (all money at \(9\%\)), they cannot quickly eliminate impossible options.

3. Not recognizing this as a range problem: Students may approach this as a system of equations problem, trying to solve for exact amounts, rather than understanding it's about determining which values fall within a feasible range of weighted averages.

Errors while executing the approach

1. Arithmetic errors in boundary calculations: When calculating maximum interest (\(\$5{,}000 \times 0.09 = \$450\)) or minimum interest (\(\$5{,}000 \times 0.06 = \$300\)), students may make basic multiplication errors, leading to incorrect boundaries and wrong elimination of answer choices.

2. Incorrect testing of specific allocations: When testing whether \(\$425\) or \(\$335\) are achievable, students may make errors in calculating interest for their chosen allocations (e.g., calculating \(\$3{,}000 \times 0.09 + \$2{,}000 \times 0.08\) incorrectly), leading them to conclude that feasible amounts are impossible.

3. Incomplete verification: Students may test only one allocation per target amount and conclude it's impossible if that specific allocation doesn't work, rather than understanding that multiple allocation combinations could achieve the target interest.

Errors while selecting the answer

1. Misreading the Roman numeral format: Students correctly identify that options II and III are possible but then select the wrong answer choice, perhaps choosing "C) I and II only" instead of "D) II and III only" due to careless reading of which Roman numerals correspond to which amounts.

2. Including eliminated options: After correctly determining that \(\$475\) is impossible due to exceeding the maximum, students may still select an answer choice that includes option I, either by forgetting their boundary analysis or second-guessing their work.