Problem 55 - Elementary and Intermediate Algebra

Question

Assume that there are no deposits or withdrawals. (IMAGE CANNOT COPY) Comparing Interest Rates. How much more interest could \(\$ 1,000\) earn in 5 years, compounded quarterly, if the annual interest rate were \(5 \frac{1}{2} \%\) instead of \(5 \% ?\)

Step-by-Step Solution

Answer

The 5.5% rate earns approximately $12.96 more in interest over 5 years.

Step 1: Understand the formula for compound interest

The compound interest formula 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 \(\$1,000\), \( r \) is the annual interest rate (as a decimal), \( n \) is the number of times interest applied per time period, and \( t \) is the time the money is invested for in years.

Step 2: Calculate interest earned with 5% rate

For a 5% annual interest rate compounded quarterly, the formula with \( P = \$1000 \), \( r = 0.05 \), \( n = 4 \), \( t = 5 \) becomes: \[ A_5 = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 5} \]Calculate \( A_5 \) to find the amount after 5 years.

Step 3: Calculate final amount for 5% rate

Compute \( A_5 = 1000 \left(1 + \frac{0.05}{4}\right)^{20} = 1000 \left(1.0125\right)^{20} \). Calculate \( (1.0125)^{20} \) and then multiply by \\(1000. \( A_5 \approx \\)1283.68 \) after 5 years.

Step 4: Calculate interest earned with 5.5% rate

For a 5.5% annual interest rate compounded quarterly, the formula is: \[ A_{5.5} = 1000 \left(1 + \frac{0.055}{4}\right)^{4 \times 5} \]Calculate \( A_{5.5} \) to find the amount after 5 years.

Step 5: Calculate final amount for 5.5% rate

Compute \( A_{5.5} = 1000 \left(1 + \frac{0.055}{4}\right)^{20} = 1000 \left(1.01375\right)^{20} \). Calculate \( (1.01375)^{20} \) and then multiply by \\(1000. \( A_{5.5} \approx \\)1296.64 \) after 5 years.

Step 6: Calculate the difference in interest earned

The additional interest earned by using the 5.5% rate is the difference \( A_{5.5} - A_5 \). Calculate \( 1296.64 - 1283.68 \). This gives approximately \$12.96.

Key Concepts

Understanding Annual Interest Rate

An annual interest rate is a percentage that represents the yearly cost of borrowing money or the yearly earnings from an investment. It is typically expressed as a percentage of the principal amount. In the context of the problem, the annual interest rates being compared are 5% and 5.5%.

When calculating compound interest, it's vital to convert the annual interest rate to a decimal. For instance, for the 5% rate, you divide 5 by 100 to get 0.05, and for the 5.5% rate, you divide 5.5 by 100 to get 0.055.

• 5% annual rate: 0.05 as a decimal
• 5.5% annual rate: 0.055 as a decimal

The annual interest rate ensures that investors understand how much they can expect their investments to grow over a year. In this scenario, the difference in rates can result in different amounts of total interest accrued over 5 years.

Deciphering the Principal Amount

The principal amount is the initial sum of money placed in investment or lent as a loan, on which interest is calculated. It's a foundational concept in both savings and loans as it determines the baseline from which interest grows.

In our problem, the principal amount is $1,000. This is the starting point of the calculation process. As a fixed amount in the calculations, it is essential for determining how much interest can be earned over time.

• Principal amount in this example: $1,000

Understanding your principal is crucial because, together with the interest rate and compounding frequency, it helps predict the future value of an investment. In such calculations, any changes in the principal amount would directly affect the calculated interest.

What Does Compounded Quarterly Mean?

Compounded quarterly refers to the frequency with which interest is calculated and added to the principal sum of an investment over time. In this scenario, it means that the interest is compounded four times a year, or every three months.

When compounding quarterly, the annual interest rate is divided by four to find the interest rate for each period. So, for example, a 5% annual interest rate becomes 1.25% per quarter, while a 5.5% annual interest rate becomes 1.375% per quarter.

• Quarterly compounding means four compounding periods per year
• Quarterly rate for 5%: 0.0125
• Quarterly rate for 5.5%: 0.01375

Compounding more frequently generally results in a higher amount of interest accrued over time compared to less frequent compounding. Thus, compounded quarterly means the investment grows faster compared to if it were compounded annually, due to the previously added interest now earning additional interest in each period.