A young executive deposits $300 at the end of each month for 4 years into an account that earns 7.2% compounded monthly. How much is in the account after the 4 years? (Round your answer to the nearest cent).
The executive then increases the deposits in order to have a total of $400,000 after 25 total years. How much additional money is still needed in order to meet the $400,000 goal? (Round your answer to the nearest cent).
How many dollars of interest will the amount in his account after 4 years make during the 4th year to the 25th year?
How much should each new deposit be in order to have a total of $400,000 after 25 total years? (Round your answer to the nearest cent).
To solve these questions, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the final amount of money after the given time period
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
Let's solve each question step by step:
1. How much is in the account after 4 years?
To find the final amount after 4 years, we need to use the given formula. Here:
P = $300 (monthly deposit)
r = 7.2% per year (as a decimal, r = 0.072)
n = 12 (compounded monthly)
t = 4 years
A = 300(1 + 0.072/12)^(12*4)
A ≈ $14,173.34 (rounded to the nearest cent)
So, there is approximately $14,173.34 in the account after 4 years.
2. How much additional money is still needed to meet the $400,000 goal?
To calculate the additional money needed, we subtract the current amount in the account from the desired goal.
Additional money needed = $400,000 - $14,173.34
Additional money needed ≈ $385,826.66 (rounded to the nearest cent)
So, an additional amount of approximately $385,826.66 is still needed to meet the $400,000 goal.
3. How many dollars of interest will the amount in the account after 4 years make during the 4th year to the 25th year?
To calculate the interest earned during the 4th year to the 25th year, we subtract the initial amount from the final amount after this time period.
Interest earned = Final amount after 25 years - Final amount after 4 years
Interest earned = $400,000 - $14,173.34
Interest earned ≈ $385,826.66 (rounded to the nearest cent)
So, the amount in the account is expected to earn approximately $385,826.66 during the 4th year to the 25th year.
4. How much should each new deposit be in order to have a total of $400,000 after 25 total years?
Now, to calculate the amount of each new deposit, we need to rearrange the compound interest formula.
A = P(1 + r/n)^(nt)
Solving for P, we get:
P = A / ((1 + r/n)^(nt))
where:
A = desired final amount ($400,000)
r = annual interest rate (as a decimal, r = 0.072)
n = number of times the interest is compounded per year (n = 12)
t = number of years (t = 25)
P = 400,000 / ((1 + 0.072/12)^(12*25))
P ≈ $463.22 (rounded to the nearest cent)
So, each new deposit should be approximately $463.22 in order to have a total of $400,000 after 25 years.