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Magazine Chapter 8: Problem 40
Find the annual percentage rate at which \(\$ 4\) will grow to \(\$ 6\) in 500 days if the interest is compounded daily. Round to the nearest tenth of a percent.
Short Answer
Expert verified
The annual percentage rate is approximately 29.0%.
Step by step solution
01
Understand the Compound Interest Formula
The formula for compound interest is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where, \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (the initial sum 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 or borrowed for, in years.
02
Identify Given Values
Extract the values from the problem statement: \( P = 4 \), \( A = 6 \), \( n = 365 \) (since the interest is compounded daily), \( t = \frac{500}{365} \).
03
Substitute Values into the Formula
Substitute the values into the compound interest formula: \[ 6 = 4 \left(1 + \frac{r}{365}\right)^{365 \times \frac{500}{365}} \]
04
Simplify the Equation
Simplify the exponent part of the equation: \[ 6 = 4 \left(1 + \frac{r}{365}\right)^{500} \]
05
Isolate the Interest Rate Term
First, divide both sides by 4 to isolate the interest rate term: \[ 1.5 = \left(1 + \frac{r}{365}\right)^{500} \]
06
Solve for the Interest Rate
Take the 500th root of both sides to solve for the term with \(r\): \[ 1.5^{\frac{1}{500}} = 1 + \frac{r}{365} \] Subtract 1 from both sides: \[ 1.0007933 - 1 = \frac{r}{365} \] \[ 0.0007933 = \frac{r}{365} \]
07
Calculate the Annual Interest Rate
Multiply both sides by 365 to isolate \(r\): \[ r = 0.0007933 \times 365 = 0.2895 \] Convert the decimal rate to a percentage: \[ r = 28.95% \] Round to the nearest tenth: \[ r \approx 29.0% \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Annual Percentage Rate (APR)
The Annual Percentage Rate (APR) is a crucial concept in finance. It represents the yearly interest generated by a sum that's charged to borrowers or paid to investors. APR gets expressed as a percentage, which helps users easily understand the annual fee they can expect on their credit or investment. In our example, we calculated the APR for an amount of money growing from \(4 to \)6 in 500 days. The interest was compounded daily. We'll explore what this means in more depth below.
Analyzing APR helps both borrowers and investors make informed financial decisions. If you're taking a loan, a higher APR means you鈥檙e paying more in interest over a year. Conversely, if you're investing, a higher APR means you鈥檙e earning more.
Analyzing APR helps both borrowers and investors make informed financial decisions. If you're taking a loan, a higher APR means you鈥檙e paying more in interest over a year. Conversely, if you're investing, a higher APR means you鈥檙e earning more.
Interest Compounded Daily
When interest is compounded daily, it means that the interest adds to the principal sum more frequently鈥�365 times a year, to be exact. Each day, the interest calculated from the day before is added to the principal, and the new total becomes the basis for the next day's interest calculation.
This mechanism harnesses the power of compound interest more effectively than less frequent compounding schedules. For example, monthly compounding adds interest 12 times a year, while daily compounding does it 365 times.
In our exercise, we used daily compounding to calculate how \(4 grew to \)6 in 500 days. Notice that more frequent compounding results in slightly higher amounts of accrued interest over the same period, compared to less frequent compounding.
Here's the mathematical representation for daily compounding:
\ \ \text{Final Amount} \ A = P \left(1 + \frac{r} {365} \right)^{365t}
In this formula, 365 represents the frequency of compounding within a year. By substituting the appropriate values, we solve for the final amount, principal, and interest rate accordingly.
This mechanism harnesses the power of compound interest more effectively than less frequent compounding schedules. For example, monthly compounding adds interest 12 times a year, while daily compounding does it 365 times.
In our exercise, we used daily compounding to calculate how \(4 grew to \)6 in 500 days. Notice that more frequent compounding results in slightly higher amounts of accrued interest over the same period, compared to less frequent compounding.
Here's the mathematical representation for daily compounding:
\ \ \text{Final Amount} \ A = P \left(1 + \frac{r} {365} \right)^{365t}
In this formula, 365 represents the frequency of compounding within a year. By substituting the appropriate values, we solve for the final amount, principal, and interest rate accordingly.
Interest Calculation
Understanding how to calculate interest, especially compound interest, is essential. Compound interest means your interest earns interest over time, unlike simple interest which is calculated only on the principal amount.
In the provided exercise, we began with a principal (\( P \)) amount of \(4 and ended up with an amount (\( A \)) of \)6 over 500 days with daily compounding. Using the compound interest formula:
\ A = P \left(1 + \frac{r}{n}\right)^{nt}\,
We then isolated our variables and solved for the annual interest rate (\( r \)):
\
In the provided exercise, we began with a principal (\( P \)) amount of \(4 and ended up with an amount (\( A \)) of \)6 over 500 days with daily compounding. Using the compound interest formula:
\ A = P \left(1 + \frac{r}{n}\right)^{nt}\,
We then isolated our variables and solved for the annual interest rate (\( r \)):
\
