Occasionally a savings account may actually pay interest compounded continuously. For each deposit, find the interest earned if interest is compounded (a) semiannually, (b) quarterly, (c) monthly, (d) daily, and (e) continuously. Use 1 year=365 days.
Principal
Rate
Time
$1006
1.6%
2 years
Question content area bottom
Part 1
(a) The interest earned if interest is compounded semiannually is $enter your response here.
(Do not round until the final answer. Then round to the nearest cent as needed.)
The formula used to calculate the interest earned for each compounding period is:
A = P(1 + r/n)^(nt) - P
Where:
A = the final amount
P = the principal
r = interest rate (in decimal form)
n = number of times interest is compounded per year
t = time in years
Given:
P = $1006
r = 1.6% or 0.016 (in decimal form)
t = 2 years
For semiannual compounding (n = 2), the calculation becomes:
A = 1006(1 + 0.016/2)^(2*2) - 1006
A = 1006(1.008)^(4) - 1006
A = 1006(1.032236864) - 1006
A = 1037.733077504 - 1006
A = 31.733077504
The interest earned if interest is compounded semiannually would be approximately $31.73.
To calculate the interest earned if interest is compounded semiannually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount (including interest)
P = Principal amount
r = Annual interest rate (expressed as a decimal)
n = Number of times interest is compounded per year
t = Number of years
Given:
P = $1006
r = 1.6% = 0.016
n = 2 (compounded semiannually)
t = 2 years
Substituting the values into the formula:
A = 1006(1 + 0.016/2)^(2*2)
A ≈ 1006(1.008)^4
A ≈ $1006 * 1.032064
A ≈ $1032.45
To find the interest earned, we subtract the principal amount from the final amount:
Interest earned = $1032.45 - $1006 = $26.45
Therefore, the interest earned if interest is compounded semiannually is $26.45.
To find the interest earned if interest is compounded semiannually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (including interest)
P = principal (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, the principal (P) is $1006, the annual interest rate (r) is 1.6% or 0.016, the number of times interest is compounded per year (n) is 2 (semiannually), and the number of years (t) is 2.
Plugging in these values into the formula, we get:
A = 1006(1 + 0.016/2)^(2*2)
A = 1006(1 + 0.008)^(4)
A = 1006(1.008)^(4)
A ≈ 1006 * 1.032032
A ≈ $1036.193872
To find the interest earned, we subtract the principal from the final amount:
Interest earned = A - P
Interest earned = $1036.193872 - $1006
Interest earned ≈ $30.193872
Therefore, the interest earned if interest is compounded semiannually is approximately $30.19.