Initial Investment??
Annual % Rate= 8
Time to Double??
Amount after 10 years= $20,000
I don't know how to figure out the initial investment and the time to double help me please?
I'll try to help. I'm not exactly sure if the money is being compounded continuously or not but I'll guide you along with the assumption that it is.
Pe^(rt) = current amount of money
P = principal
e = natural base = about 2.71
r = rate (.08 in this case)
t = time... in years
(^ is exponentiation by the way)
Since we know the amount after 10 years = $20000, we can easily solve for P
P = 20000/(e^(.08*10)) - about $9009.
For the time to double. Imagine that you initialy invested $1. Just one into the bank account.
The amount you want eventually is $2, right? (Duh, it has to double.)
2 = 1(e^(.08)*t)
Take the natural log of both sides. ln. It gets rid of the e.
ln 2 = ln (e^(.08 *t))
Based on properties of log and natural log, the .08*t is left behind on the right.
ln 2 = .08*t
t = ln 2/.08 = about 8.66 years
(For doubling problems, you can always check your answers by using the rule of 72. For instance, since the rate is 8%, divide 72 by that. You should get 9. That's kinda close to the time you got, right? Don't use this as a replacement for doing the actual work, but if you're ever unsure, you can use this method to check.) Hope this helped. Peace out.
thank you XD that helped alot!!
Gladly. Peace out.
To calculate the initial investment and time to double, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Amount after time t
P = Principal or initial investment
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Time in years
Since we are given the amount after 10 years (A = $20,000) and the annual interest rate (r = 8%), we can rewrite the formula as:
$20,000 = P(1 + 0.08/n)^(n*10)
Now, we need to find the values of P (initial investment) and n (compounding frequency) to solve the equation.
To find the initial investment (P), we can use algebra to isolate it:
P = $20,000 / (1 + 0.08/n)^(n*10)
Next, let's determine the compounding frequency (n). This information is usually given, so if it's not provided, we can assume it to be either annually (n = 1), semi-annually (n = 2), quarterly (n = 4), or monthly (n = 12). The more compounding periods within a year, the faster the investment will grow.
Now, we can calculate the time to double using the formula:
Time to double = log(2) / log(1 + r/n)
Substituting the given values into the equation:
Time to double = log(2) / log(1 + 0.08/n)
By solving this equation, you can find the time it takes for the investment to double.