at what rate of interest per annum,the amount of 10000 will be doubled in 5 years,interest being compounded quarterly
To determine the rate of interest per annum at which the amount of 10,000 will be doubled in 5 years with quarterly compounding, we can use the compound interest formula:
A = P(1 + r/n)^(n*t)
Where:
A is the future value or final amount
P is the principal amount (initial investment)
r is the interest rate per period (unknown)
n is the number of compounding periods per year (quarterly compounding)
t is the number of years
Given:
P = 10,000
A = 2 * P = 20,000
n = 4 (quarterly compounding)
t = 5 years
Rearranging the formula, we can solve for r:
(1 + r/n)^(n*t) = A/P
Substituting the given values:
(1 + r/4)^(4*5) = 20,000/10,000
Simplifying further:
(1 + r/4)^20 = 2
Now, we need to solve for (1 + r/4)^20, which can be done by applying the concept of logarithms:
log[(1 + r/4)^20] = log(2)
Using the property of logarithms, we can bring the exponent down:
20 * log(1 + r/4) = log(2)
Dividing both sides by 20:
log(1 + r/4) = log(2) / 20
Since log(1 + r/4) is the power that 10 must be raised to, we can rewrite the equation:
10^(log(1 + r/4)) = 10^(log(2) / 20)
Simplifying further:
1 + r/4 = 10^(log(2) / 20)
Subtracting 1 from both sides:
r/4 = 10^(log(2) / 20) - 1
Multiplying both sides by 4:
r = 4 * (10^(log(2) / 20) - 1)
Using a calculator, we can evaluate this expression to find the value of r.