at what rate of interest per annum,the amount of 10000 will be doubled in 5 years,interest being compounded quarterly
let the quarterly rate be j
1000(1+j)^20 = 2000
(1+j)^20 = 2
1+j = 2^(1/20) = 1.035265
j = .035265
the annual rate compounded quarterly is .14106
= 14.1 %
(good luck finding that these days)
no problem. go to any bookie or loan shark!
To determine the rate of interest per annum required for an amount to double in a given time period, compounded quarterly, we can use the compound interest formula.
The formula for compound interest with quarterly compounding is:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (twice the initial amount in this case)
P = the principal amount (initial amount)
r = the annual interest rate (unknown)
n = the number of times that interest is compounded per year (4, since it's quarterly)
t = the number of years
In this case, we want the initial amount of $10,000 to double in 5 years. So we have:
A = 2P (since we want the final amount to be twice the initial amount)
P = $10,000
n = 4
t = 5
Plugging these values into the formula, we get:
2P = P(1 + r/4)^(4*5)
Canceling out the P on both sides, we have:
2 = (1 + r/4)^(20)
To solve for r, we need to isolate it. One way to do this is by using logarithms:
log(2) = log((1 + r/4)^(20))
Using the property of logarithms, we can bring the exponent down:
log(2) = 20log(1 + r/4)
Now we can solve for r by isolating it:
log(1 + r/4) = log(2)/20
To remove the logarithm, we can raise both sides as a power of 10:
10^(log(1 + r/4)) = 10^(log(2)/20)
This simplifies to:
1 + r/4 = 10^(log(2)/20)
Now we can isolate r:
r/4 = 10^(log(2)/20) - 1
Multiply both sides by 4 to get:
r = 4(10^(log(2)/20) - 1)
Using a calculator, we can evaluate the right side of the equation to get the approximate value for r.