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Chapter 8: Problem 19

How much money should be deposited today in an account that earns \(9.5 \%\) compounded monthly so that it will accumulate to \(\$ 10,000\) in three years?

Short Answer

Expert verified
If the result is computed correctly, it should be approximately \$7381.31. This means if \$7381.31 is deposited today, it will accumulate to \$10,000 in three years with a monthly compounded interest rate of 9.5%.

Step by step solution

01

Identify Given Values

The given values in this problem are the interest rate (\(9.5\%\) or 0.095 in decimal form), compounding frequency (monthly or 12 times a year), time period (3 years), and future value (\$10,000). We aim to find the present value, i.e., the initial deposit that should be made today.
02

Use the Compound Interest Formula

The compound interest formula is \(A = P(1 + r/n)^(nt)\), where 'A' is the amount of money accumulated after 'n' years, including interest. 'P' is the principal amount (the initial amount of money), 'r' is the annual interest rate in decimal form, 'n' is the number of times that interest is compounded per year, and 't' is the time the money is invested for in years.
03

Rearrange the Compound Interest Formula

To find the initial deposit (or 'P'), we would rearrange the formula to \(P = A/((1 + r/n)^(nt))\).
04

Substitute the Known Values and Compute

Substitute \(A = $10,000\), \(r = 0.095\), \(n = 12\), and \(t = 3\) into the formula, and compute for 'P'.

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Most popular questions from this chapter

Describe how to determine what you can afford for your monthly mortgage payment.

In terms of paying less in interest, which is more economical for a $$ 150,000\( mortgage: a 30 -year fixed-rate at \)8 \%\( or a 20 -year fixed-rate at \)7.5 \%$ ? How much is saved in interest?

In Exercises 11-18, a. Determine the periodic deposit. Round up to the nearest dollar. b. How much of the financial goal comes from deposits and how much comes from interest? $$ \begin{array}{|l|l|l|l|} \hline \$ \text { ? at the end of each month } & 7.25 \% \text { compounded monthly } & 40 \text { years } & \$ 1,000,000 \\ \hline \end{array} $$

Suppose your credit card has a balance of \(\$ 3600\) and an annual interest rate of \(16.5 \%\). You decide to pay off the balance over two years. If there are no further purchases charged to the card, a. How much must you pay each month? b. How much total interest will you pay?

Solve for \(P\) : $$ A=\frac{P\left[(1+r)^{t}-1\right]}{r} . $$ What does the resulting formula describe?
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