6.how long will it take for a sum of money to quadruple at 10 percent compounded annually ?
7. At what rate compounded semi-annually will RM2000 become RM 3500 in five years ?
1.1^n = 4
n log 1.1 = log 4
n = log 4/log 1.1 = 14.5
(1 + r/2)^10 = 3500/2000
10 log (1 + r/2) = log 1.75
log (1 + r/2) = .0243
1 + r/2 = 1.05756
r/2 = .05756
r = .115 or 11.5 %
To calculate the time it takes for a sum of money to quadruple with compound interest, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount (quadruple the initial sum)
P = principal amount (initial sum)
r = annual interest rate (10% in this case)
n = number of times the interest is compounded per year (1 for annually)
t = number of years
Let's calculate the time it takes for the money to quadruple in each scenario.
6. To quadruple a sum of money at 10% compounded annually:
We need to find the value of t in the equation:
4P = P(1 + 0.10/1)^(1*t)
Simplifying the equation, we have:
4 = (1 + 0.10)^t
Now, we need to solve for t. Taking the logarithm base 10 on both sides:
log(4) = log((1 + 0.10)^t)
log(4) = t * log(1.10)
t = log(4) / log(1.10)
Calculating this using a calculator, we find:
t ≈ 11.4509
Therefore, it will take approximately 11.4509 years for a sum of money to quadruple at 10% compounded annually.
7. To find the rate compounded semi-annually:
We need to find the value of r in the equation:
A = P(1 + r/2)^(2*5)
3500 = 2000(1 + r/2)^(10)
Let's solve for r. Taking the 10th root of both sides:
(3500/2000)^(1/10) = 1 + r/2
Now, subtracting 1 from both sides:
(3500/2000)^(1/10) - 1 = r/2
Multiplying both sides by 2:
2 * [(3500/2000)^(1/10) - 1] = r
Calculating this using a calculator, we find:
r ≈ 0.0977 or approximately 9.77%
Therefore, a rate of approximately 9.77% compounded semi-annually will cause RM2000 to become RM3500 in five years