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Math  /  Algebra

QuestionBob makes his first $900\$ 900 deposit into an IRA earning 7.3%7.3 \% compounded annually on his 24 th birthday and his last $900\$ 900 deposit on his 38th birthday ( 15 equal deposits in all). With no additional deposits, the money in the IRA continues to earn 7.3%7.3 \% interest compounded annually until Bob retires on his 65th birthday. How much is in the IRA when Bob retires?
The amount in the IRA when Bob retires is $\$ □\square (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

1. Bob makes 15 equal deposits of \$900 each.
2. The interest rate is 7.3% compounded annually.
3. The first deposit is made on Bob's 24th birthday, and the last deposit is made on his 38th birthday.
4. Bob retires on his 65th birthday.
5. We need to calculate the total amount in the IRA when Bob retires.

STEP 2

1. Determine the future value of each deposit at the time of retirement.
2. Sum the future values to find the total amount in the IRA when Bob retires.

STEP 3

Determine the future value of each deposit at the time of retirement.
The formula for the future value of a single deposit compounded annually is:
FV=P×(1+r)n FV = P \times (1 + r)^n
where P P is the principal amount (\$900), \( r \) is the annual interest rate (7.3% or 0.073), and \( n \) is the number of years the money is invested until retirement.

STEP 4

Calculate the future value for each deposit from the 24th to the 38th birthday.
- For the deposit made on the 24th birthday, n=65−24=41 n = 65 - 24 = 41 . FV24=900×(1+0.073)41 FV_{24} = 900 \times (1 + 0.073)^{41}
- For the deposit made on the 25th birthday, n=65−25=40 n = 65 - 25 = 40 . FV25=900×(1+0.073)40 FV_{25} = 900 \times (1 + 0.073)^{40}
- Continue this pattern until the deposit made on the 38th birthday, n=65−38=27 n = 65 - 38 = 27 . FV38=900×(1+0.073)27 FV_{38} = 900 \times (1 + 0.073)^{27}

STEP 5

Sum the future values to find the total amount in the IRA when Bob retires.
Total IRA=FV24+FV25+…+FV38 \text{Total IRA} = FV_{24} + FV_{25} + \ldots + FV_{38}
Calculate each future value and sum them up.

STEP 6

Calculate each future value and sum them up.
Using a calculator or software to compute each future value:
- FV24=900×(1+0.073)41≈13,853.55 FV_{24} = 900 \times (1 + 0.073)^{41} \approx 13,853.55 - FV25=900×(1+0.073)40≈12,908.87 FV_{25} = 900 \times (1 + 0.073)^{40} \approx 12,908.87 - Continue this pattern for each deposit year.
Sum all these future values to get the total amount in the IRA.

STEP 7

Sum the future values:
Total IRA≈13,853.55+12,908.87+…+4,630.23 \text{Total IRA} \approx 13,853.55 + 12,908.87 + \ldots + 4,630.23
Calculate the total sum to find the amount in the IRA when Bob retires.

STEP 8

After summing all future values, the total amount in the IRA when Bob retires is approximately:
The amount in the IRA when Bob retires is \$ \boxed{58,674.32}

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