If $10,000 is invested at x percent simple annual interest for n years, which of the following represents the total...

1. Translate the problem requirements: Understand what "simple annual interest" means and identify that we need the total interest earned (not the total amount), expressed in terms of principal (\(\$10,000\)), rate (x percent), and time (n years)
2. Apply simple interest logic: Recall that simple interest means interest is calculated only on the original principal each year, so annual interest = principal × rate, and total interest = annual interest × number of years
3. Convert percentage to decimal form: Transform the x percent rate into mathematical form \(\frac{\mathrm{x}}{100}\) to make it usable in calculations
4. Construct the final expression: Combine all components (principal, converted rate, and time) to create the formula for total interest earned

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is asking in plain English.

We have \(\$10,000\) invested at "x percent simple annual interest" for "n years". The key word here is "simple" interest - this means the interest is calculated only on the original amount each year, not on any accumulated interest.

The question asks for the "total amount of interest" earned - not the final value of the investment, but just the interest portion.

Let's think about this with a concrete example: If I invest \(\$10,000\) at \(5\%\) simple interest for 3 years, how much interest do I earn each year? Each year, I earn \(5\%\) of the original \(\$10,000\), which is \(\$500\). After 3 years, my total interest is \(3 \times \$500 = \$1,500\).

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

2. Apply simple interest logic

Now let's build the logic step by step using our understanding.

With simple interest:

• Year 1: Interest = \(5\%\) of \(\$10,000 = \$500\)
• Year 2: Interest = \(5\%\) of \(\$10,000 = \$500\) (still calculated on original amount)
• Year 3: Interest = \(5\%\) of \(\$10,000 = \$500\)

Total interest over 3 years = \(\$500 + \$500 + \$500 = 3 \times \$500 = \$1,500\)

In general terms:

• Annual interest = Rate × Principal
• Total interest = Annual interest × Number of years
• Total interest = Rate × Principal × Number of years

So for our problem:
Total interest = \(\mathrm{x}\% \times \$10,000 \times \mathrm{n \text{ years}}\)

3. Convert percentage to decimal form

Now we need to handle the "x percent" properly in mathematical form.

When we say "x percent," we mean \(\frac{\mathrm{x}}{100}\) in decimal form.
For example:

• \(5\% = \frac{5}{100} = 0.05\)
• \(12\% = \frac{12}{100} = 0.12\)

So "x percent" = \(\frac{\mathrm{x}}{100}\)

Substituting this into our formula:
Total interest = \(\left(\frac{\mathrm{x}}{100}\right) \times \$10,000 \times \mathrm{n}\)

4. Construct the final expression

Let's put all the pieces together:

Total interest = \(\left(\frac{\mathrm{x}}{100}\right) \times \$10,000 \times \mathrm{n}\)

We can rearrange this as:
Total interest = \(\$10,000 \times \mathrm{n} \times \left(\frac{\mathrm{x}}{100}\right)\)

Written in mathematical notation: \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\)

Let's verify this makes sense with our concrete example:

• Principal = \(\$10,000\)
• Rate = \(5\% = \frac{5}{100}\)
• Time = 3 years
• Total interest = \(10,000 \times 3 \times \left(\frac{5}{100}\right) = 30,000 \times \left(\frac{5}{100}\right) = 30,000 \times 0.05 = \$1,500\) ✓

This matches our earlier calculation!

4. Final Answer

The total amount of interest earned is represented by: \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\)

Looking at the answer choices, this matches choice C: \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\)

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Confusing Simple Interest with Compound Interest

Students often misread the problem and assume compound interest instead of simple interest. This leads them to think the interest compounds annually, making them look for formulas with exponential terms like \(\left(1+\frac{\mathrm{x}}{100}\right)^\mathrm{n}\), when simple interest means interest is calculated only on the original principal each year.

2. Misunderstanding what the question is asking for

The question asks for the "total amount of interest" earned, but students may confuse this with the "total amount" (principal + interest) after n years. This confusion leads them to look for formulas that represent the final value of the investment rather than just the interest portion.

3. Not recognizing the percentage conversion requirement

Students may not immediately recognize that "x percent" needs to be converted to decimal form \(\left(\frac{\mathrm{x}}{100}\right)\) for mathematical calculations. They might try to work with "x" directly, not realizing it represents a percentage that must be divided by 100.

Errors while executing the approach

1. Incorrect handling of the percentage conversion

Even when students understand they need to convert percentage to decimal, they may place the conversion incorrectly in the formula or forget to include it entirely, leading to expressions like \(10,000\mathrm{nx}\) instead of \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\).

2. Misplacing variables in the multiplication

Students may correctly identify all components (principal, rate, time) but arrange them incorrectly in the final expression, such as writing \(10,000\mathrm{x}\left(\frac{\mathrm{n}}{100}\right)\) instead of \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\), confusing which variable gets divided by 100.

Errors while selecting the answer

1. Choosing compound interest formulas

After working through the problem, students may still select answer choices D or E that contain \(\left(1+\frac{\mathrm{x}}{100}\right)^\mathrm{n}\) terms, which are compound interest formulas, because these "look more complex" and students assume complex problems require complex formulas.

2. Selecting the wrong mathematical notation

Students may arrive at the correct conceptual answer but choose a wrong option due to misreading the mathematical notation, such as confusing \(10,000\left(\frac{\mathrm{x}}{100}\right)^\mathrm{n}\) with \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\), not carefully noting the placement of the exponent n.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient smart numbers
Let's select values that will make our calculations clean:
• Principal = \(\$10,000\) (given)
• Interest rate: \(\mathrm{x} = 5\%\) (a common, easy-to-work-with percentage)
• Time period: \(\mathrm{n} = 3\) years (a manageable number for calculations)

Step 2: Calculate the actual interest using simple interest logic
With simple interest, we earn the same amount each year based only on the original principal:
• Annual interest = \(\$10,000 \times 5\% = \$10,000 \times 0.05 = \$500\)
• Total interest over 3 years = \(\$500 \times 3 = \$1,500\)

Step 3: Test each answer choice with our smart numbers
Substitute \(\mathrm{x} = 5\) and \(\mathrm{n} = 3\) into each option:

Choice A: \(10,000(\mathrm{x}^\mathrm{n}) = 10,000(5^3) = 10,000(125) = 1,250,000\) ✗

Choice B: \(10,000\left(\frac{\mathrm{x}}{100}\right)^\mathrm{n} = 10,000\left(\frac{5}{100}\right)^3 = 10,000(0.05)^3 = 10,000(0.000125) = 1.25\) ✗

Choice C: \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right) = 10,000(3)\left(\frac{5}{100}\right) = 10,000(3)(0.05) = 1,500\) ✓

Choice D: \(10,000\left(1+\frac{\mathrm{x}}{100}\right)^\mathrm{n} = 10,000(1+0.05)^3 = 10,000(1.05)^3 = 10,000(1.157625) = 11,576.25\) ✗

Choice E: \(10,000\mathrm{n}\left(1+\frac{\mathrm{x}}{100}\right) = 10,000(3)(1.05) = 31,500\) ✗

Step 4: Verify our answer
Only Choice C gives us \(\$1,500\), which matches our calculated simple interest. This confirms that the correct formula for simple interest is: Principal × Rate (as decimal) × Time.