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Chapter 4: Problem 72

A sum of \(\$ 1000\) was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to \(\$ 1435.77\) in the given time, what was the interest rate?

Short Answer

Expert verified
The interest rate was approximately 9.21%.

Step by step solution

01

Understanding Compound Interest Formula

The formula for calculating compound interest is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \(A\) is the amount after time \(t\), \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time in years.
02

Identify Known Values

From the problem statement, we have \(P = 1000\), \(A = 1435.77\), \(t = 4\), and \(n = 2\) since the interest is compounded semiannually.
03

Substitute Known Values into the Formula

Substitute the known values into the compound interest formula: \[ 1435.77 = 1000 \left(1 + \frac{r}{2}\right)^{8} \] (since \(nt = 2 \times 4 = 8\)).
04

Solve for \( \left(1 + \frac{r}{2}\right)^8 \)

Divide both sides of the equation by 1000 to isolate the power: \[ \left(1 + \frac{r}{2}\right)^8 = \frac{1435.77}{1000} \approx 1.43577 \]
05

Taking the 8th Root

Take the 8th root of both sides to eliminate the exponent: \[ 1 + \frac{r}{2} = (1.43577)^{1/8} \] Calculate this to find the result about 1.04606.
06

Solve for the Interest Rate \(r\)

Subtract 1 from both sides to isolate \(\frac{r}{2}\): \[ \frac{r}{2} = 1.04606 - 1 = 0.04606 \]. Multiply both sides by 2 to solve for \(r\): \[ r = 0.09212 \]. Convert this to a percentage to find the annual rate: \( r \approx 9.21\% \).

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Key Concepts

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

Compound Interest Formula
The compound interest formula is a powerful tool used to calculate the growth of an investment over time. Unlike simple interest, which is calculated on the initial principal alone, compound interest takes into account the accumulated interest from previous periods. Here is the formula:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:
  • \(A\) is the future value of the investment, including interest
  • \(P\) is the principal investment amount
  • \(r\) is the annual interest rate (decimal)
  • \(n\) is the number of times interest is compounded per year
  • \(t\) is the time the money is invested or borrowed for, in years
This formula allows you to determine how much an investment will grow over time given different compounding periods and interest rates. Understanding each component of the formula is essential for solving problems related to compound interest.
Annual Interest Rate
The annual interest rate is a key component in understanding the growth of investments. It represents the percentage of the principal earned or paid in interest over one year. When using the compound interest formula, the annual interest rate, \(r\), must be converted into a decimal. For example, a 5% annual interest rate becomes 0.05.In the original problem, we find this rate by rearranging the compound interest formula to solve for \(r\). This requires inputting all known values and isolating the variable we want to find. The annual rate expresses how much the investment grows in a year before considering compounding effects.
Semiannual Compounding
Semiannual compounding refers to compounding the interest twice a year. It means the interest gets calculated and added to the principal every six months. This affects the number of compounding periods in the formula, \(n\), which becomes 2 for semiannual compounding.In our exercise, semiannual compounding is considered by substituting \(n = 2\) into the compound interest formula. As a result, the total number of compounding periods, \(nt\), becomes 8, since it triplicates over 4 years. Each period, the interest applies not only to the initial principal but also to the interest accumulated in previous periods, making it crucial to consider the frequency of compounding to achieve accurate results.
Solving for Interest Rate
When given the compound interest equation, solving for the interest rate \(r\) involves algebraic manipulation. Initially, substitute all known values into the formula except for \(r\). For our exercise:\[ 1435.77 = 1000 \left(1 + \frac{r}{2}\right)^8 \]The next step is to simplify the equation:
  • Divide both sides by 1000 to isolate the base
\[ \left(1 + \frac{r}{2}\right)^8 = 1.43577 \]To remove the exponent, take the 8th root of both sides:\[ 1 + \frac{r}{2} = (1.43577)^{1/8} \]Subtract 1 and multiply by 2 to isolate \(r\):\[ r = 0.09212 \]Finally, convert this decimal into a percentage for clarity, resulting in an annual interest rate of approximately 9.21%. This process exemplifies how algebraic skills and understanding compounded functions can help determine unknowns in financial equations.

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