Question

Find the sum accumulated by $1000 in 87days at 5% by (i) exact and (ii) banker’s interest method

Answer 1

To calculate the accumulated sum using both the exact interest and the banker’s interest methods, we will use the following formulas:

1. **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.

2. **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
1. **Exact Interest Accumulation**: \( A \approx 1011.92 \)
2. **Banker's Interest Accumulation**: \( A \approx 1012.08 \)

Thus, the final accumulated sums are:
- **Exact Interest**: \( \$1011.92 \)
- **Banker's Interest**: \( \$1012.08 \)