What is the sum accumulated by $1000 in 85 days at 5% by (i) exact and (ii) banker's interest method
To calculate the sum accumulated by $1000 at 5% interest for a period of 85 days, we need to apply different formulas for exact interest and banker’s interest calculations. For both methods, interest is typically expressed on an annual basis, which means we have to adjust for the fraction of the year that 85 days represents.
(i) **Exact Interest Method:**
Exact interest uses a calendar year of 365 days to calculate interest for the exact number of days an amount is borrowed or lent. The formula for calculating exact interest (I) is:
\[I = P \times r \times \frac{t}{365}\]
Where:
- \(P\) is the principal amount ($1000)
- \(r\) is the annual interest rate (5% or 0.05 as a decimal)
- \(t\) is the time the money is borrowed or invested, in days (85 days)
Plugging in the numbers, we get:
\[I = $1000 \times 0.05 \times \frac{85}{365}\]
\[I = $11.64\]
The sum accumulated is the principal plus the interest, so:
\[Sum = P + I\]
\[Sum = $1000 + $11.64\]
\[Sum = $1011.64\]
(ii) **Banker’s Interest Method (Ordinary Interest):**
Banker’s interest uses a year of 360 days (which simplifies the arithmetic for calculations by assuming each month has exactly 30 days). The formula for calculating banker’s interest (I) is similar:
\[I = P \times r \times \frac{t}{360}\]
Let's do the calculation:
\[I = $1000 \times 0.05 \times \frac{85}{360}\]
\[I = $11.81\]
The sum accumulated under banker’s interest then would be:
\[Sum = P + I\]
\[Sum = $1000 + $11.81\]
\[Sum = $1011.81\]
It's important to note that these calculations are simplified and assume simple interest rather than compound interest, which could be a different scenario where interest is compounded periodically within the year.