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 $32.5832.58.
(Do not round until the final answer. Then round to the nearest cent as needed.)
Part 2
(b) The interest earned if interest is compounded quarterly is $enter your response here.
(Do not round until the final answer. Then round to the nearest cent as needed.)
To calculate the interest earned for each case, we'll use the formula for compound interest:
A = P(1 + r/n)^(nt) - P
Where:
A = the final amount (including principal and interest)
P = the principal (initial deposit)
r = the annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = the number of years
Given:
P = $1006
r = 1.6% = 0.016
t = 2 years
(a) Semiannually (n = 2):
A = 1006(1 + 0.016/2)^(2*2) - 1006
= 1006(1.008)^4 - 1006
= 1006(1.0326) - 1006
= $32.58 (rounded to the nearest cent)
(b) Quarterly (n = 4):
A = 1006(1 + 0.016/4)^(4*2) - 1006
= 1006(1.004)^8 - 1006
= 1006(1.0321) - 1006
= $32.11 (rounded to the nearest cent)
(c) Monthly (n = 12):
A = 1006(1 + 0.016/12)^(12*2) - 1006
= 1006(1.0013)^24 - 1006
= 1006(1.0315) - 1006
= $31.53 (rounded to the nearest cent)
(d) Daily (n = 365):
A = 1006(1 + 0.016/365)^(365*2) - 1006
= 1006(1.000043)^730 - 1006
= 1006(1.03125) - 1006
= $31.25 (rounded to the nearest cent)
(e) Continuously:
A = P * e^(rt)
= 1006 * e^(0.016*2) - 1006
= 1006 * e^(0.032) - 1006
= 1006 * 1.0325 - 1006
= $32.50 (rounded to the nearest cent)
Therefore, the interest earned if interest is compounded (b) quarterly is $32.11.
To calculate the interest earned for each compounding period, you can use the formula:
A = P * (1 + r/n)^(n*t)
Where:
A is the final amount (including principal and interest)
P is the principal (initial deposit)
r is the interest rate (in decimal form)
n is the number of times compounded per year
t is the time in years
Given the information you provided:
P = $1006
r = 1.6% or 0.016
t = 2 years
Part 1: Compounded semiannually (n = 2)
A = 1006 * (1 + 0.016/2)^(2*2)
A = 1006 * (1 + 0.008)^4
A ≈ 1006 * 1.03258
A ≈ $1032.58
Interest earned = A - P
Interest earned = $1032.58 - $1006
Interest earned ≈ $32.58
Part 2: Compounded quarterly (n = 4)
Similar to the above calculation, you can use the same formula and substitute n with 4.
A = 1006 * (1 + 0.016/4)^(4*2)
A = 1006 * (1 + 0.004)^8
You can complete this calculation to find the interest earned.
To find the interest earned for each compounding period, we can use the formula for compound interest:
A = P * (1 + r/n)^(n*t)
where:
A = the final amount (including principal and interest)
P = principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
Given:
Principal (P) = $1006
Annual interest rate (r) = 1.6% = 0.016
Time (t) = 2 years
Now, let's calculate the interest earned for each compounding period:
(a) Semiannually (n = 2):
A = $1006 * (1 + 0.016/2)^(2*2) = $1038.11630
Interest earned = A - P = $1038.11630 - $1006 = $32.11630
Rounding to the nearest cent, the interest earned is $32.12.
(b) Quarterly (n = 4):
A = $1006 * (1 + 0.016/4)^(4*2) = $1038.6753
Interest earned = A - P = $1038.6753 - $1006 = $32.6753
Rounding to the nearest cent, the interest earned is $32.68.
(c) Monthly (n = 12):
A = $1006 * (1 + 0.016/12)^(12*2) = $1038.8948
Interest earned = A - P = $1038.8948 - $1006 = $32.8948
Rounding to the nearest cent, the interest earned is $32.89.
(d) Daily (n = 365):
A = $1006 * (1 + 0.016/365)^(365*2) = $1038.9201
Interest earned = A - P = $1038.9201 - $1006 = $32.9201
Rounding to the nearest cent, the interest earned is $32.92.
(e) Continuously:
A = P * e^(r*t) (where e is a mathematical constant approximately equal to 2.71828)
A = $1006 * e^(0.016*2) = $1038.9306
Interest earned = A - P = $1038.9306 - $1006 = $32.9306
Rounding to the nearest cent, the interest earned is $32.93.
Therefore, the interest earned for each compounding period is:
(a) Semiannually: $32.12
(b) Quarterly: $32.68
(c) Monthly: $32.89
(d) Daily: $32.92
(e) Continuously: $32.93