Question

Sophie deposited money into an account in which interest is compoundedsemiannually at a rate of 3.4%. She made no other deposits or withdrawals and thetotal amount in her account after 13 years was $24,155.71. How much did shedeposit? Round answer to nearest whole number.

Answer 1

The compound interest formula for the amount of money A in an account after t years if a principal P is invested at an annual interest rate r (written as a decimal) which is compounded n times a year, is:

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

In this case, the interest rate of 3.4% (0.034), the amount after 13 years, $24,155.71 and the number of compounding periods a year, 2, are given. The principal P is unknown.

Then, isolate P from the equation:

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

Replace A=24,155.71, r=0.034, n=2 and t=13 to find the principal P that Sophie deposited at the beginning:

`\begin{gathered} P=24155.71*(1+(0.034)/(2))^(-2*13) \\ =15,583.85975... \\ \approx15,584 \end{gathered}`

Therefore, to the nearest whole number, the amount that Sophie deposited was $15,584.