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Nathan took out a student loan for \(\$1200\) at \(10\%\) annual interest, compounded annually. If he did not repay any of the loan or interest during the first \(3\) years, which of the following is the closest to the amount of interest he owed for the \(3\) years.
Let's understand what's happening here in everyday terms. Nathan borrowed money - specifically $1200 - and the bank charges him 10% interest each year. The key detail is that this interest is "compounded annually," which means each year the interest gets added to what he owes, and then next year's interest is calculated on this new, larger amount.
Think of it like this: if you owe $100 and the interest is 10%, you don't just owe $10 in interest each year. In year 1, you owe $10 interest. But in year 2, you owe 10% of $110 (the original $100 plus the $10 interest from year 1).
What we need to find: The total amount of interest Nathan owes after 3 years (not the total he owes, just the interest portion).
Process Skill: TRANSLATE - Converting the compound interest language into a step-by-step growth pattern
Let's follow Nathan's loan balance year by year, calculating how much he owes at the end of each year.
Starting amount: $1200
End of Year 1:
Interest for this year = 10% of $1200 = \(0.10 \times \$1200 = \$120\)
Total owed = \(\$1200 + \$120 = \$1320\)
End of Year 2:
Now we calculate 10% of the NEW balance ($1320)
Interest for this year = 10% of $1320 = \(0.10 \times \$1320 = \$132\)
Total owed = \(\$1320 + \$132 = \$1452\)
End of Year 3:
Again, we calculate 10% of the current balance ($1452)
Interest for this year = 10% of $1452 = \(0.10 \times \$1452 = \$145.20\)
Total owed = \(\$1452 + \$145.20 = \$1597.20\)
Notice how each year's interest gets bigger because it's calculated on a larger and larger amount - this is the power of compound interest!
To find the total interest Nathan owes, we need to subtract what he originally borrowed from what he owes now.
Total amount owed after 3 years: $1597.20
Original loan amount: $1200
Total interest owed = \(\$1597.20 - \$1200 = \$397.20\)
Let's double-check this by adding up the interest from each year:
Year 1 interest: $120
Year 2 interest: $132
Year 3 interest: $145.20
Total: \(\$120 + \$132 + \$145.20 = \$397.20\) ✓
Our calculated interest is $397.20. Looking at the answer choices:
$397.20 is closest to $400, which is choice C.
Process Skill: CONSIDER ALL CASES - Since the question asks for the "closest" amount, we need to compare our exact calculation to all given options
The closest amount to the interest Nathan owed for 3 years is $400 (Choice C).
Our calculation showed exactly $397.20 in interest, and among the given choices, $400 is the nearest value.
1. Confusing compound interest with simple interest: Students often misunderstand "compounded annually" and calculate simple interest instead. They might think: "10% for 3 years = 30% total interest" and calculate \(\$1200 \times 0.30 = \$360\). This leads directly to answer choice A, but ignores the compounding effect where each year's interest is added to the principal for the next year's calculation.
2. Misinterpreting what the question asks for: The question asks for "the amount of interest he owed for the 3 years" - just the interest portion. Some students might think they need to find the total amount owed (principal + interest) rather than just the interest component, which could lead them to look for answers around $1597 instead of $397.
1. Calculation errors in year-by-year tracking: When manually calculating each year's compound interest, students often make arithmetic mistakes. For example, they might incorrectly calculate 10% of $1320 as $130 instead of $132, or make errors when adding the interest to get the new balance each year.
2. Forgetting to update the principal each year: Even when students understand compound interest conceptually, they might calculate Year 2 interest as 10% of the original $1200 instead of 10% of the new balance $1320. This systematic error would significantly underestimate the total interest.
3. Using the compound interest formula incorrectly: Students who try to use the formula \(\mathrm{A} = \mathrm{P}(1 + \mathrm{r})^\mathrm{t}\) might make errors such as forgetting to subtract the principal at the end, or miscalculating \((1.1)^3\), leading to incorrect final amounts.
1. Not recognizing this is an approximation question: Students might calculate the exact interest as $397.20 but then look for this exact value among the choices. Since none of the choices match exactly, they might panic or choose randomly instead of selecting the closest option (C. $400).
2. Selecting based on first glance rather than careful comparison: When comparing $397.20 to the choices, students might quickly pick B. $390 because it "starts with 39" like their answer, without carefully noting that $400 is actually closer (difference of $2.80 vs $7.20).