How much money should be deposited today in an account that earns 4% compounded monthly so that it will accumulate to 9000 in three years?
To find the initial deposit that should be made today in order to accumulate to $9000 in three years at an interest rate of 4% compounded monthly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($9000)
P = the principal or initial deposit we want to find
r = the interest rate per period (4% or 0.04)
n = the number of compounding periods per year (monthly compounding, so 12)
t = the number of years (3)
Plugging in the values:
9000 = P(1 + 0.04/12)^(12*3)
We can simplify this equation:
9000 = P(1.003333...)^(36)
Now, divide both sides of the equation by (1.003333...)^(36):
P = 9000 / (1.003333...)^(36)
Using a calculator, we can find that (1.003333...)^(36) is approximately 1.1255. Therefore:
P = 9000 / 1.1255
P ≈ 7996.90
Therefore, an initial deposit of approximately $7996.90 should be made today in order to accumulate to $9000 in three years at an interest rate of 4% compounded monthly.
To find out how much money should be deposited today, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Accumulated amount
P = Initial deposit
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case, we want to find the initial deposit (P), so we rearrange the formula:
P = A / (1 + r/n)^(nt)
Now, let's plug in the given values:
A = 9000
r = 0.04 (4% as a decimal)
n = 12 (compounded monthly)
t = 3 years
P = 9000 / (1 + 0.04/12)^(12*3)
Now, let's calculate the value:
P = 9000 / (1 + 0.003333)^36
P = 9000 / (1.003333)^36
P = 9000 / 1.127465
P ≈ 7980.99
Therefore, an initial deposit of approximately $7980.99 should be made today in order to accumulate $9000 in three years with a 4% annual interest rate compounded monthly.
To calculate the initial deposit needed to accumulate a specific amount over a certain period of time with compound interest, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial deposit
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
In this case, we're given that the future value (A) is $9,000, the annual interest rate (r) is 4% (or 0.04 in decimal form), the interest is compounded monthly (n = 12), and the investment period (t) is 3 years.
Plugging in these values into the formula, we get:
9000 = P(1 + 0.04/12)^(12*3)
Simplifying the equation:
9000 = P(1.003333)^36
Now, divide both sides of the equation by (1.003333)^36:
9000 / (1.003333^36) = P
Using a calculator, we can find that (1.003333)^36 ≈ 1.125899.
Dividing 9000 by 1.125899:
7965.56 ≈ P
Therefore, approximately $7,965.56 should be deposited today in order to accumulate $9,000 in three years with a 4% annual interest rate compounded monthly.