Question 1176170
Not sure what "decompressed" means here so will just ignore it
:
If $ 500 in an account paying 8.5% annual interest, compounded semiannually, how long will it take for the account to increase to $ 1000?
:
Using the formula {{{A = P(1+(r/n))^(tn)}}} where:
A = resulting amt after t time
P = initial amt
r = interest rate in decimal form
n = no. of compounding times per yr
t = no. of yrs
:
{{{500(1+(.085/2))^(2t) = 1000}}}
simplify, divide both sides by 500
{{{(1+(.085/2))^(2t) = 2}}}
{{{(1.0425)^(2t) = 2}}}
using natural logs
2t*ln(1.0425) = ln(2)
:
2t = {{{ln(2)/ln(1.0425)}}}
using your calc
2t = 16.6535
divide by 2
t = 8.327 ~ 8.5 yrs to reach $1000 in the account
:
:
 How long will it take for the account to increase to $ 41000 if compounded continuously?
{{{500(1+(.085/2))^(2t) = 41000}}}
simplify, divide both sides by 500
{{{(1+(.085/2))^(2t) = 82}}}
{{{(1.0425)^(2t) = 82}}}
using natural logs
2t*ln(1.0425) = ln(82)
2t = {{{ln(82)/ln(1.0425)}}}
using your calc
2t = 105.8756
t = 52.938 ~ 53 yrs to reach $41000 in the account