Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

Interest on savings An amount of money \(P\) is deposited in a savings account that pays interest at a rate of \(r\) percent per year compounded quarterly; the principal and accumulated interest are left in the account. Find a formula for the total amount in the account after \(n\) years.

Step-by-Step Solution

Answer

The formula is \( A = P \left(1 + \frac{r}{400}\right)^{4n} \).

Step 1: Understand Quarterly Compounding

In quarterly compounding, the interest is compounded four times a year. Hence, the interest rate per compounding period is \( \frac{r}{4} \) percent.

Step 2: Calculate the Number of Compounding Periods

The total number of compounding periods over \( n \) years is \( 4n \), since interest is compounded quarterly.

Step 3: Convert Percent to Decimal Format

To perform calculations, convert the interest rate into decimal form: \( \text{rate per period} = \frac{r}{100 \times 4} \) or \( \frac{r}{400} \).

Step 4: Use the Compound Interest Formula

The formula for compound interest when the compounding is at regular intervals is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \). Here, \( n = 4 \), and the formula becomes: \[ A = P \left(1 + \frac{r}{400}\right)^{4n} \]

Step 5: Write Down the Final Formula

Substitute the given values and steps into the compound interest formula: \[ A = P \left(1 + \frac{r}{400}\right)^{4n} \] This formula gives the total amount in the account after \( n \) years with interest compounded quarterly at rate \( r \% \).

Key Concepts

Quarterly Compounding

When savings in an account are compounded quarterly, it simply means that the interest is calculated and added to the account balance four times a year. This is different from annual compounding, where interest is only calculated once in a year. This frequent addition of interest means that the account grows faster compared to less frequent compounding. Each quarter, new interest is calculated not only on the original principal but also on any interest that has already been added to the account. So, every three months, the balance is recalculated, and interest is applied on the latest total, effectively increasing the balance more than it would otherwise. Quarterly compounding uses the formula:\[ A = P \left(1 + \frac{r}{400}\right)^{4n} \]where:

• \(P\) is the principal amount.
• \(r\) is the annual interest rate, divided by 4 to reflect quarterly compounding.
• \(n\) is the number of years.

This quarterly compounding concept helps in growing the savings significantly over time.

Interest Rate Conversion

Interest rates are usually provided on an annual basis, known as the annual percentage rate (APR). When dealing with compound interest, it is crucial to convert this annual rate into a rate that reflects the compounding period being used, such as quarterly. To convert the annual interest rate for quarterly compounding, you divide the annual rate by the number of quarters in a year, which is four. So, if the annual rate is \( r \% \), the interest per quarter is \( \frac{r}{4} \% \). For calculation purposes, also convert the percentage to a decimal by dividing the rate per quarter by 100. This gives:\[ \text{Rate per quarter (as a decimal)} = \frac{r}{400} \]. This step is crucial because it lets us use the rate appropriately in the compound interest formula, helping to accurately determine the accumulated amount over time.

Number of Compounding Periods

The number of compounding periods is the total number of times compounding occurs over the entire term of the investment or loan. For quarterly compounding, the interest is added four times each year. Thus, if you want to find the number of compounding periods over a certain number of years, you multiply the number of years by four. For instance, if you plan to keep your money in a savings account for \( n \) years, the number of compounding periods will be:\[ 4n \] This calculation is important because it reflects how frequently interest is compounded and thereby affects the total interest earned. A greater number of compounding periods means more frequent interest calculations, which contributes to the growth of the account balance faster than with annual compounding. Understanding this concept allows you to maximize your returns effectively over time.