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If \(\$10,000\) is invested at \(\mathrm{x}\) percent simple annual interest for \(\mathrm{n}\) years, which of the following represents the total amount of interest, in dollars, that will be earned by this investment in the \(\mathrm{n}\) years?
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
Now let's build the logic step by step using our understanding.
With simple interest:
Total interest over 3 years = \(\$500 + \$500 + \$500 = 3 \times \$500 = \$1,500\)
In general terms:
So for our problem:
Total interest = \(\mathrm{x}\% \times \$10,000 \times \mathrm{n \text{ years}}\)
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:
So "x percent" = \(\frac{\mathrm{x}}{100}\)
Substituting this into our formula:
Total interest = \(\left(\frac{\mathrm{x}}{100}\right) \times \$10,000 \times \mathrm{n}\)
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:
This matches our earlier calculation!
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.
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.
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.
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.
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 \(10,000\mathrm{n}\left(\frac{\mathrm{x}}{100}\right)\).
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.
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.
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.
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.