Question

bob makes his first $700 deposit into an ira earning 6.7% compounded annually on his 24th birthday and his last $700 deposit on his 35th birthday (12 equal deposits in all). with no additional deposits, the money in the ira continues to earn 6.7% 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.)

Explanation:

Step1: Calculate the future - value of the annuity

The formula for the future - value of an ordinary annuity is \(F = A\times\frac{(1 + r)^{n}-1}{r}\), where \(A=\$700\), \(r = 0.067\), and \(n = 12\).

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First, calculate \((1 + 0.067)^{12}\approx2.1579\). Then \((1 + 0.067)^{12}-1\approx1.1579\). And \(\frac{(1 + 0.067)^{12}-1}{0.067}\approx17.2821\). So \(F = 700\times17.2821=\$12097.47\)

Step2: Calculate the future - value of the single amount

This amount \(F=\$12097.47\) will be compounded for \(t=65 - 35=30\) years. The compound - interest formula is \(A = P(1 + r)^{t}\), where \(P = 12097.47\), \(r=0.067\), and \(t = 30\)

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Calculate \((1+0.067)^{30}\approx7.1079\). Then \(A=12097.47\times7.1079\approx\$85974.27\)

Answer:

\(85974.27\)