if naomi wants to accumulate 1,000,000 by investing money every year into her savings account at 3% for 30 years until retirement,how much does she need to deposit each year
P+Int.=1000000. P=Principal=Total Dep.
P+(0.03)(30)P=1000000
P+0.9P=1000000.
P=526315.79=Total dep./30years
Dep./yr=526315.79/30=17543.81
To calculate how much Naomi needs to deposit each year to accumulate 1,000,000 in her savings account after 30 years at an interest rate of 3%, we can use the formula for the future value of an ordinary annuity:
Future Value = Payment x [(1 + Interest Rate)^Number of Periods - 1] / Interest Rate
In this case:
- Future Value = 1,000,000
- Interest Rate = 3% = 0.03 (decimal form)
- Number of Periods = 30 (since she will be depositing each year for 30 years)
Let's plug in the values and solve for Payment:
1,000,000 = Payment x [(1 + 0.03)^30 - 1] / 0.03
First, let's simplify the equation by calculating (1 + 0.03)^30:
(1 + 0.03)^30 ≈ 2.4272462
Now, let's substitute this value back into the equation:
1,000,000 = Payment x (2.4272462 - 1) / 0.03
1,000,000 = Payment x 1.4272462 / 0.03
Divide both sides of the equation by 1.4272462 / 0.03:
Payment = 1,000,000 / (1.4272462 / 0.03)
Payment ≈ 20,898.58
Therefore, Naomi needs to deposit approximately $20,898.58 each year to accumulate 1,000,000 in 30 years at a 3% interest rate.