how long will it take for the investment to double at an interest rate 1.7% if the interest is compunded quarterly?hint : you my assume that $100 were invested
let y = years
2 = [1 + (.017 / 4)]^(y / 4)
y / 4 = log(2) / log[1 + (.017 / 4)]
is the answer 163 years?
To find out how long it will take for the investment to double, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (double the initial investment in this case)
P = the principal amount (the initial investment of $100)
r = the annual interest rate as a decimal (1.7% as 0.017)
n = the number of times interest is compounded per year (quarterly, so 4)
t = the number of years
Let's solve for t:
200 = 100(1 + 0.017/4)^(4t)
To isolate t, we divide both sides by 100:
2 = (1.00425)^(4t)
Now, we can take the logarithm of both sides with any base. Let's use the natural logarithm (ln):
ln(2) = ln(1.00425)^(4t)
Using the logarithmic property, we can bring the exponent down:
ln(2) = (4t)ln(1.00425)
Now, divide both sides by 4ln(1.00425):
t = ln(2) / (4ln(1.00425))
Using a calculator, we find t to be approximately 40.4815 quarters. Since each year has 4 quarters, we can convert this to years by dividing by 4:
t = 40.4815 / 4
Therefore, it will take approximately 10.12 years for the investment to double with a 1.7% interest rate compounded quarterly.