Invest $2800 @ 4.7% a year (compound). How long it will take to double the amount invested to the nearest tenth of the year?
so the question is when is interest compounded? annually, daily, monthly? every 10th of the year? It matters.
assuming annual compounding, you want t such that
1.047^t = 2
To determine how long it will take for an investment to double at a given interest rate, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount after compounding
P = the original principal (amount invested)
r = the annual interest rate (expressed as a decimal)
n = the number of times compounding occurs per year
t = the number of years
In this case, the principal (P) is $2800, the annual interest rate (r) is 4.7% (or 0.047 as a decimal), and we want to find the time it takes to double the investment, so the final amount (A) will be $5600 (twice the principal).
Substituting these values into the formula, we get:
5600 = 2800(1 + 0.047/n)^(nt)
To solve this equation for t, we can use logarithms. Take the natural logarithm (ln) of both sides of the equation:
ln(5600) = ln(2800(1 + 0.047/n)^(nt))
Now, use logarithmic properties to break down the equation:
ln(5600) = ln(2800) + ln(1 + 0.047/n)^(nt)
Using the power rule of logarithms, we can bring down the exponent:
ln(5600) = ln(2800) + nt * ln(1 + 0.047/n)
Divide both sides of the equation by ln(1 + 0.047/n):
(ln(5600) - ln(2800)) / ln(1 + 0.047/n) = nt
Finally, divide both sides of the equation by n and solve for t:
t = (ln(5600) - ln(2800)) / (n * ln(1 + 0.047/n))
Since we want to find the time to the nearest tenth of a year, we can substitute different values for n (number of compounding periods per year) and calculate t using a calculator until we find a value that results in the investment doubling.
For example, let's start with n = 1 (which means compounding annually):
t = (ln(5600) - ln(2800)) / (1 * ln(1 + 0.047/1))
t ≈ 14.5 years
So, it would take approximately 14.5 years to double the investment if the compounding occurs annually. You can repeat this process for different values of n (such as n = 2 for semi-annual compounding or n = 12 for monthly compounding) to get more accurate results.