compound math question?

Question

can you help me w/this math hmk problem:

If $500 is deposited in an account paying 8.5% annual interest, how long will it take for the account to increase to $800? Compound semiannually? Compound continuously?

Answer 1

Compound semiannually:
800 = 500(1+0.085/2)^(2t)
t = 0.5ln(8/5)/ln(1+0.085/2) = 5.65 years

Compound continuously:
800 = 500e^0.085t
t = ln(8/5)/0.085 = 5.53 years

Answer 2

The compound interest formula is

p = a(1 + i)^t

Where p is final amount, a is beginning amount
i = interest rate and t is the number of periods

compounded annually

800 = 500(1.085)^t

1.6 = (1.085)^t

t Ln1.085 = Ln 1.6

t = Ln 1.6/Ln1.085

t = 5.76 yrs (compounded annually)

compounded semi-annually

800 = 500(1 + 0.085/2)^(2t)

1.6 = (1 + 0.085/2)^(2t)

1.6 = (1.0425)^(2t)

2t = Ln 1.6/Ln 1.0425

t = 5.65 yrs (compounded semi-annually)

compounded continuously

800 = 500(e)^(0.085t)

1.6 = (e)^(0.085t)

since Ln e = 1

0.085t = Ln 1.6

t = 5.53 yrs (compounded continuously)
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