at what rate of interest a certain sum will be doubled in 8 year?
To find the rate of interest at which a certain sum will be doubled in 8 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount (twice the initial amount)
P = Principal (initial amount)
r = Rate of interest (unknown)
n = Number of compounding periods per year (assuming it is once per year)
t = Time in years (8 years in this case)
We need to solve this equation for r.
Step 1: Rearranging the equation
Divide both sides of the equation by P to eliminate it from the equation:
(1 + r/n)^(nt) = 2
Step 2: Substitute known values
Substitute the known values into the equation:
(1 + r/1)^(8*1) = 2
Step 3: Simplifying the equation
Simplify the exponent:
(1 + r)^(8) = 2
Step 4: Isolating the variable
Take the logarithm of both sides of the equation to isolate the variable r:
log((1 + r)^(8)) = log(2)
Step 5: Apply logarithm properties
Using the logarithmic identity log(b^c) = c*log(b), we can bring down the exponent:
8*log(1 + r) = log(2)
Step 6: Divide both sides of the equation
Divide both sides of the equation by 8:
log(1 + r) = log(2) / 8
Step 7: Take the exponent
Apply the exponential function e^(log(x)) = x to both sides of the equation:
1 + r = e^(log(2) / 8)
Step 8: Subtract 1 from both sides
Subtract 1 from both sides of the equation:
r = e^(log(2) / 8) - 1
Using a calculator or software, you can approximate the value of r.