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Chapter 5: Problem 14

Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. \(P=\$ 2500, r=9 \%, t=10 \frac{1}{2}\), compounded semiannually

Short Answer

Expert verified
The accumulated amount after \(10\frac{1}{2}\) years, when a principal amount of \(\$2500\) is invested at an interest rate of \(9\%\) per year compounded semiannually, is approximately \(\$5154.87\).

Step by step solution

01

Convert the given values to appropriate units

Before plugging the given values into our compound interest formula, we need to convert them to appropriate units. 1. Convert the annual interest rate to a decimal: \(r = 9\% = 0.09\) 2. Express the number of years as a decimal: \(t = 10\frac{1}{2} = 10.5\) Now that we have the given values in the appropriate units, we can proceed with the calculation.
02

Plug the values into the compound interest formula

Using the given values, we can plug them into the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\) Since the investment is compounded semiannually, there are \(n = 2\) compounding periods per year. Therefore, the formula becomes: \(A = 2500(1 + \frac{0.09}{2})^{2(10.5)}\)
03

Calculate the accumulated amount

Now that we have the formula set up with the given values, we can calculate the accumulated amount: \(A = 2500(1 + \frac{0.09}{2})^{2(10.5)} \approx 2500(1.045)^{21} \approx \$5154.87\)
04

State the final answer

The accumulated amount after \(10\frac{1}{2}\) years, when a principal amount of \(\$2500\) is invested at an interest rate of \(9\%\) per year compounded semiannually, is approximately \(\$5154.87\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Accumulated Amount
The accumulated amount in any investment refers to the total sum of money that has grown over a specified period. This includes both the original principal sum and the interest that has been accumulated over time. In the context of the given exercise, the accumulated amount is what you find at the end of the investment term.
  • The formula to calculate the accumulated amount with compound interest is given by: \( A = P(1 + \frac{r}{n})^{nt} \).
  • Here, \(A\) represents the accumulated amount.
  • Total amount is achieved by adding the interest accrued to the original principal.
In our problem, the accumulated amount after investing the principal at an annual interest rate for the specified number of years leads us to a final value of approximately \( \$5154.87 \). This signifies successful growth of the initial investment over time through compound interest.
Interest Rate
The interest rate is a crucial factor in determining how much an investment grows. It is essentially the cost of borrowing money or the return on investment expected by the lender or investor.
  • Expressed as a percentage, it represents the proportion of the principal that is paid as interest over a specific period.
  • In this exercise, the interest rate is \(9\%\), which is converted to a decimal for calculations, resulting in \( r = 0.09 \).
A higher interest rate typically means higher returns, as the principal grows more each compounding period. However, the interest rate is applied to the principal in smaller increments when compounding more frequently, as seen with semiannual compounding in this exercise.
Semiannual Compounding
Semiannual compounding means that the interest is calculated and added to the principal twice a year, every six months. This type of compounding affects how fast your investment grows over time.
  • When interest is compounded semiannually, the number of compounding periods per year \(n\) is \(2\).
  • The formula for compound interest in semiannual terms becomes: \( A = P(1 + \frac{r}{2})^{2t} \).
This method accelerates the accumulation of interest because each increase in principal grows at an ever-accelerating rate due to the frequent compounding. In the exercise, semiannual compounding makes a significant difference by boosting the total accumulated amount.
Principal Investment
The principal investment is the initial sum of money put into the investment. It is critical to understanding how your money is working for you.
  • The principal investment, denoted as \(P\), represents the starting amount before any interest is earned.
  • In the exercise, the principal is given as \( \$2500 \).
The strength of your initial principal can greatly impact the final accumulated amount. A larger principal directly results in higher interest in each compounding period, underlining its importance in any financial growth calculation. The principal serves as the base from which your total earnings stem.

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Most popular questions from this chapter

Trust FunDs Carl is the beneficiary of a \(\$ 20,000\) trust fund set up for him by his grandparents. Under the terms of the trust, he is to receive the money over a 5 -yr period in equal installments at the end of each year. If the fund earns interest at the rate of \(9 \% /\) year compounded annually, what amount will he receive each year?

Determine which of the sequences are geometric progressions. For each geometric progression, find the seventh term and the sum of the first seven terms. $$ 0.004,0.04,0.4,4, \ldots $$

FINANCING A CAR The price of a new car is \(\$ 16,000\). Assume that an individual makes a down payment of \(25 \%\) toward the purchase of the car and secures financing for the balance at the rate of \(10 \% /\) year compounded monthly. a. What monthly payment will she be required to make if the car is financed over a period of \(36 \mathrm{mo}\) ? Over a period of \(48 \mathrm{mo}\) ? b. What will the interest charges be if she elects the 36 -mo plan? The 48 -mo plan?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the future value of an annuity consisting of \(n\) payments of \(R\) dollars each-paid at the end of each investment period into an account that earns interest at the rate of \(i\) per period-is \(S\) dollars, then $$ R=\frac{i S}{(1+i)^{n}-1} $$

FINANCING A HomE Eight years ago, Kim secured a bank loan of \(\$ 180,000\) to help finance the purchase of a house. The mortgage was for a term of \(30 \mathrm{yr}\), with an interest rate of \(9.5 \% /\) year compounded monthly on the unpaid balance to be amortized through monthly payments. What is the outstanding principal on Kim's house now?
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