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.
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.)
1 answer