A person desires to create a fund to be invested at 10% compound interest per annum to provide for a price of 300 every year. The amount we should invest is:
To determine the amount that needs to be invested, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the desired annual payment of 300
P = the principal amount (the amount to be invested)
r = the interest rate per annum (10% or 0.1)
n = the number of times interest is compounded per year (considering it compounds annually, n = 1)
t = the number of years
We want to solve for P, so we rearrange the formula:
P = A / (1 + r/n)^(nt)
Plugging in the given values:
A = 300
r = 0.1
n = 1
P = 300 / (1 + 0.1/1)^(1*t)
Since there is no specific time mentioned, we can assume it is invested for an indefinite period. Thus, t goes to infinity.
As t approaches infinity, (1 + 0.1/1)^(1*t) becomes (1 + 0.1)^(1*t) = (1.1)^(1*t) = (1.1)^t.
The investment will only be enough to provide an annual payment of 300 if the interest keeps growing at the same rate.
So, for the amount to be invested, we require P = (A / (1 + r/n)^(nt)) = (300 / (1.1)^t).
Note: Without specifying a certain time period, it is not possible to calculate the exact amount to be invested.
To determine the amount that needs to be invested in order to provide for a price of 300 every year at a compound interest rate of 10% per annum, you can use the formula for calculating the present value of an annuity.
The formula for the present value of an annuity is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value (the amount to be invested)
PMT = Payment per period (300 in this case)
r = Interest rate per period (0.10 in this case)
n = Number of periods (1 year in this case)
Let's substitute the values into the formula and calculate the present value:
PV = 300 * (1 - (1 + 0.10)^(-1)) / 0.10
PV = 300 * (1 - (1.10)^(-1)) / 0.10
PV = 300 * (1 - 0.9091) / 0.10
PV = 300 * 0.0909 / 0.10
PV = 27.27
Therefore, the amount that needs to be invested is approximately 27.27.
To determine the amount that needs to be invested, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal (initial amount)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, the desired annual price that needs to be provided is $300, and the interest rate is 10% per annum. So, we need to solve for the principal amount (P).
Let's substitute the known values:
A = $300
r = 10% = 0.10
t = 1 year (as we need to provide $300 every year)
n = 1 (annual compounding)
Now, rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Substituting the values:
P = $300 / (1 + 0.10/1)^(1 * 1)
P = $300 / (1 + 0.10)^1
P = $300 / (1.10)^1
P = $300 / 1.10
P ≈ $272.73
Therefore, the amount that should be invested is approximately $272.73 to provide $300 every year at a compound interest rate of 10% per annum.