Determine the principal P that must be invested at rate
r = 3
1
2
%,
compounded monthly, so that $700,000 will be available for retirement in
t = 13
years. (Round your answer to the nearest cent.)
P = $
Well, well, well, looks like someone's planning for retirement! So, you want to know how much money you need to invest. Alright, let's do some calculations here.
We have the interest rate, r = 3.125%. Now, let's convert that into decimal form, shall we? So, r = 0.03125.
We also have the time period, t = 13 years.
Now, we need to find the principal, P, which is the amount of money you need to invest.
The formula to find the future value of an investment is given by:
FV = P(1 + r/n)^(nt)
In our case, we want to find P. So, let's rearrange the formula:
P = FV / (1 + r/n)^(nt)
Plugging in the values, we get:
P = $700,000 / (1 + 0.03125/12)^(12 * 13)
Calculating that, the principal P that you need to invest comes out to be approximately $322,239.02.
So, grab your piggy bank and start saving, my friend! Retirement is calling!
To determine the principal amount that must be invested, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = final amount ($700,000 in this case)
P = principal amount
r = annual interest rate (3.5%)
n = number of compounding periods per year (12, since it's compounded monthly)
t = number of years (13)
Substituting the given values into the formula, we can solve for P:
$700,000 = P(1 + 0.035/12)^(12*13)
$700,000 = P(1 + 0.002917)^156
$700,000 = P(1.002917)^156
Now we can solve for P:
To determine the principal P that must be invested, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount accumulated after t years
P = the principal (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, we want to find the principal P needed to accumulate $700,000 in 13 years at an annual interest rate of r = 3(1/2)%, compounded monthly. We can substitute the given values into the formula and solve for P.
r = 3(1/2)% = 3/2/100 = 0.015 (as a decimal)
n = 12 (since interest is compounded monthly)
t = 13
A = $700,000
Now let's plug in the values into the formula:
$700,000 = P(1 + 0.015/12)^(12*13)
Now we can solve for P. We divide both sides of the equation by (1 + 0.015/12)^(12*13) to isolate P:
P = $700,000 / (1 + 0.015/12)^(12*13)
Using a calculator, we can evaluate the expression on the right-hand side and round our final answer to the nearest cent:
P ≈ $313,882.91
Therefore, the principal P that must be invested at a rate of 3(1/2)%, compounded monthly, to accumulate $700,000 in 13 years is approximately $313,882.91.
Do you really mean 312% 1nt. rate?
If this is an error, you can recalculate using the correct %. I'll
use 31.2%.
Pt = Po(1+r)^n = $700,000.
r = (31.2%/12) / 100%=0.026 = Monthly % rate expressed as a decimal.
n = 12 comp./yr * 13yrs = 156 = The #
of compounding periods.
Po(1.026)^156 = 700,000.
Po = 700000 / (1.026)^156 = $12,767.61.