Answer :
To determine which type of savings account would earn the most interest, we need to understand the difference between simple interest and compound interest.
Simple interest is calculated on the original principal only. It means that you earn interest each period on the initial amount you invested, without taking into consideration any interest that has previously accrued. The formula for simple interest is:
\[ I = P \cdot r \cdot t \]
Where:
- \( I \) is the interest earned
- \( P \) is the principal amount (the initial amount of money)
- \( r \) is the annual interest rate (in decimal form)
- \( t \) is the time the money is invested for, in years
Compound interest, on the other hand, is calculated on the principal and on any interest that has previously been earned. So each period, you earn interest on the previous period's total. This is because the interest gets added to the principal, so you effectively earn "interest on the interest." Over time, compound interest results in exponential growth of your investment. The formula for compound interest, compounded annually, is:
\[ A = P \cdot (1 + r)^t \]
Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest
- \( P \) is the principal amount (the initial amount of money)
- \( r \) is the annual interest rate (in decimal form)
- \( t \) is the time the money is invested for, in years
Now, let's use these formulas to determine which option would earn the most interest with an initial investment of $10,000 and an interest rate of 6% over a period of 1 year.
**Option A: 6% Simple Interest**
\[ I = 10000 \cdot 0.06 \cdot 1 = 600 \]
The total value with simple interest after one year would be:
\[ Total = P + I = 10000 + 600 = 10600 \]
**Option B: 6% Compound Interest (Incorrectly labeled as linear)**
\[ A = 10000 \cdot (1 + 0.06)^1 = 10000 \cdot 1.06 = 10600 \]
This option has the same result after one year because compound interest has not yet started to have a significant effect; it only diverges from simple interest after more than one compounding period.
**Option C: 6% Compound Interest (Correctly labeled as exponential)**
With the same calculation as in option B (since it's also compound interest), the amount after one year would be:
\[ A = 10000 \cdot 1.06 = 10600 \]
**Option D: 6% Simple Interest (Incorrectly labeled as exponential)**
\[ I = 10000 \cdot 0.06 \cdot 1 = 600 \]
Total with simple interest after one year would be the same as Option A:
\[ Total = 10000 + 600 = 10600 \]
In this specific instance, since we're only looking at a time period of one year, the simple interest and compound interest options both give the same result of $10,600. However, over multiple years, compound interest would result in a higher balance due to exponential growth, since it builds interest upon the interest that has previously been earned.
Therefore, the correct answer is:
C. 6% compound interest, because the account has exponential growth.
