Question

1. At the age of 33, to save for retirement, you decide to deposit $60 at the end of each month in an IRA that pays 5.5% compounded monthly.

Determine how much you will have in the IRA when you retire at age 65 and find the interest.
2. you would like to have $5000 in 5 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that paus 4.5% compounded semiannually.
Determine how much you should deposit at the end of every six months.
Also how much of the $5,000 comes from deposits and how much comes from interest

Answer 1

To determine how much you will have in the IRA at retirement and the interest earned, use the formula for compound interest. Substituting the values gives a value of approximately $44,153.46. The interest earned is approximately $22,453.46.

Step-by-step explanation:

To determine how much you will have in the IRA when you retire at age 65, you can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

• A is the amount in the IRA at age 65
• P is the monthly deposit ($60)
• r is the annual interest rate (5.5% or 0.055)
• n is the number of times interest is compounded per year (12 for monthly)
• t is the number of years (65 - 33 = 32)

Substituting these values into the formula gives:

A = 60(1 + 0.055/12)^(12*32)

A ≈ $44,153.46

To find the interest earned, subtract the total amount deposited from the amount in the IRA:

Interest = A - (P * 12 * t)

Interest = $44,153.46 - (60 * 12 * 32) = $22,453.46

Answer 2

The future value of an annuity formula is used to determine the amount in the IRA at retirement after making monthly deposits and the amount that should be deposited semiannually to reach a goal of $5,000 for a vacation, as well as interest earned in both scenarios.

Step-by-step explanation:

To determine how much you will have in the IRA when you retire at age 65 after starting at age 33 with a monthly deposit of $60 and an annual interest rate of 5.5% compounded monthly, we need to use the future value of an annuity formula:

FV = P × [( (1 + r)n - 1 ) / r ]

where:
FV = future value of the annuity (amount in the IRA at retirement)
P = monthly payment ($60)
r = monthly interest rate (5.5% annual rate / 12 months = 0.0045833)
n = total number of payments (32 years × 12 months/year = 384 payments)

Therefore, the future value of the annuity is:

FV = 60 × [[(1 + 0.0045833)384 - 1] / 0.0045833]

We calculate the future value and then subtract the total deposits from it to find the total interest earned by retirement.

For the second question, to have $5,000 in 5 years with deposits made semiannually at an interest rate of 4.5% compounded semiannually, we use the future value of an annuity formula again but solve for the periodic payment P.

The formula rearranges to:

P = FV / [( (1 + r)n - 1 ) / r ]

where: FV = $5,000, r = semiannual interest rate (4.5% annual rate / 2), and n = number of deposits (5 years × 2 deposits/year).

After finding P, we calculate total deposits by multiplying P by n, and then we subtract total deposits from $5,000 to determine how much comes from interest.