Answer:
Bob and Alice want to remodel their bathroom in 4 years. They estimate the job will cost $35,000. How much must they invest now at an annual interest rate of 4% compounded quarterly to achieve their goal?
we need to use the present value formula:
present value = future value / (1 + interest rate)ⁿ = $35,000 / (1 + 1%)¹⁶ = $29,848.74
The Morenos invest $9000 in an account that grows to $11,000 in 4 years. What is the annual interest rate r if interest is compounded
a. Quarterly
$9,000 = $11,000 / (1 + r)¹⁶
(1 + r)¹⁶ = $11,000 / $9,000 = 1.2222
¹⁶√(1 + r) = ¹⁶√1.2222
1 + r = 1.01262
r = 0.01262 = 1.26% ⇒ quarterly interest rate
annual interest rate = 1.26% x 4 = 5.048%
b. Continuously
future value = present value x eᵃⁿ
$11,000 = $9,000 x 2.718⁴ⁿ
2.718⁴ⁿ = $9,000 / $11,000 = 0.818181818
⁴√2.718⁴ⁿ = ⁴√0.818181818
2.718ⁿ = 0.95107
nlog2.718 = log0.95107
n 0.434249452 = -0.021787543
n = -0.021787543 / 0.434249452 = -0.05017, since n must be positive, then
n = 0.05017 = 5.017%
D) a. 5.048% b. 5.017%
