how long will it take a sum of money to tripe itself at (i) 9.5% compound interest and (ii) 9.5% simple interest?
3=1(1.095)^t
assuming compounding annually.
ln3=t*ln(1.095) solve for t.
ii) when does t*(1.095)=2 ?
12.2
To calculate the time it takes for a sum of money to triple itself at compound interest and simple interest, we need to use the respective formulas for compound interest and simple interest.
(i) Compound Interest:
The formula for compound interest is given by:
A = P*(1 + r/n)^(n*t)
Where:
A = Final amount
P = Principal amount (initial sum of money)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Time (in years)
In this case, we want to find the time it takes for the principal to triple itself, so A = 3P.
3P = P*(1 + r/n)^(n*t)
We need to rearrange this equation to solve for time (t).
(ii) Simple Interest:
The formula for simple interest is given by:
A = P*(1 + r*t)
Where:
A = Final amount
P = Principal amount (initial sum of money)
r = Annual interest rate (in decimal form)
t = Time (in years)
In this case, we want to find the time it takes for the principal to triple itself, so A = 3P.
3P = P*(1 + r*t)
We need to rearrange this equation to solve for time (t).
Let's solve for time in both cases:
(i) Compound Interest:
3P = P*(1 + r/n)^(n*t)
Divide both sides by P to simplify:
3 = (1 + r/n)^(n*t)
Now, take the natural logarithm (ln) of both sides:
ln(3) = ln[(1 + r/n)^(n*t)]
Next, use the property of logarithms to bring down the exponent:
ln(3) = (n*t) * ln(1 + r/n)
Rearrange the equation to solve for time (t):
t = [ln(3)] / [(n * ln(1 + r/n))]
(ii) Simple Interest:
3P = P*(1 + r*t)
Divide both sides by P to simplify:
3 = 1 + r*t
Subtract 1 from both sides:
2 = r*t
Rearrange the equation to solve for time (t):
t = 2 / r
By plugging in the values for the annual interest rate (9.5% = 0.095) and the compounding frequency (n), you can calculate the time it will take for the sum of money to triple itself in each case.