To calculate the accumulated sum using both the exact interest and the banker’s interest methods, we will use the following formulas:
-
Exact interest formula: \[ A = P \left(1 + rt\right) \] where
- \( A \) is the total amount accumulated,
- \( P \) is the principal amount (initial investment),
- \( r \) is the rate of interest (as a decimal),
- \( t \) is the time in years.
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Banker's interest formula (assuming a year consists of 360 days): \[ A = P \left(1 + rt\right) \] but here \( t \) will be calculated using 360 days as the year length.
Step 1: Exact Interest Calculation
- Given:
- Principal \( P = 1000 \)
- Rate \( r = 5% = 0.05 \)
- Time in days \( t = 87 \left( \frac{87}{365} \text{ years} \right) \)
Calculating \( t \): \[ t = \frac{87}{365} \approx 0.238356 \]
Now substituting values into the formula for exact interest: \[ A = 1000 \left(1 + 0.05 \cdot 0.238356\right) \] \[ A \approx 1000 \left(1 + 0.0119178\right) \] \[ A \approx 1000 \cdot 1.0119178 \approx 1011.92 \]
Step 2: Banker’s Interest Calculation
In the banker’s method, we use 360 days for the calculations.
- Given:
- Principal \( P = 1000 \)
- Rate \( r = 5% = 0.05 \)
- Time in days is still \( 87 \) days.
Calculating \( t \): \[ t = \frac{87}{360} \approx 0.2416667 \]
Now substituting values into the banker’s interest formula: \[ A = 1000 \left(1 + 0.05 \cdot \frac{87}{360}\right) \] \[ A = 1000 \left(1 + 0.05 \cdot 0.2416667\right) \] \[ A = 1000 \left(1 + 0.0120833\right) \] \[ A \approx 1000 \cdot 1.0120833 \approx 1012.08 \]
Final Results
- Exact Interest Accumulation: \( A \approx 1011.92 \)
- Banker's Interest Accumulation: \( A \approx 1012.08 \)
Thus, the final accumulated sums are:
- Exact Interest: \( $1011.92 \)
- Banker's Interest: \( $1012.08 \)
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