Question

Applying Economic Concepts Suppose that you deposit \(\$ 100\) into your savings account, which earns 3 percent interest per year. Use what you've learned about calculating interest to determine how much money you'll have in your account at the end of one year and at the end of six years.

Short answer

After one year, you'll have \$103; after six years, approximately \$119.41.

Step-by-step solution

Understand Simple and Compound Interest

Interest can be calculated using simple interest or compound interest. Simple interest is calculated on the principal amount only, while compound interest is calculated on the principal plus any interest accrued over previous periods.

Choose the Correct Formula for Interest Calculation

In this exercise, it's reasonable to assume that the interest is compounded annually. Therefore, we'll use the compound interest formula: \[ A = P (1 + r)^n \]where \( A \) is the amount of money accumulated after \( n \) years, including interest, \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate (in decimal), and \( n \) is the number of years.

Calculate the Amount After One Year

Substitute the values into the compound interest formula for one year:\[ P = 100, \ r = 0.03, \ n = 1 \]The calculation becomes:\[ A = 100 (1 + 0.03)^1 = 100 \times 1.03 = 103 \]At the end of one year, you'll have \( \$103 \) in your account.

Calculate the Amount After Six Years

Substitute the values into the compound interest formula for six years:\[ P = 100, \ r = 0.03, \ n = 6 \]The calculation becomes:\[ A = 100 (1 + 0.03)^6 \approx 100 \times 1.194054 = 119.41 \]At the end of six years, you'll have approximately \( \$119.41 \) in your account.

Key concepts

Simple Interest

Simple interest is one of the basic ways to calculate the interest earned on a principal amount. This method calculates interest based solely on the original amount of money deposited or loaned, called the principal, without considering any previous interest accrued. The formula to compute simple interest is given by:

• \( I = P \times r \times n \)

where \( I \) represents the interest earned, \( P \) is the principal amount, \( r \) is the rate of interest per period (expressed as a decimal), and \( n \) is the number of periods.
Simple interest is straightforward. It makes it easy to predict the interest returned over time since the amount added each period remains constant. However, simple interest does not leverage the potential benefits of compounding, where previously earned interest itself generates more interest.

Interest Rate

The interest rate is a percentage that indicates how much interest is payable on the principal amount over a set period. It effectively determines the return on an investment or the cost of borrowing money and is typically expressed on an annual basis known as the Annual Percentage Rate (APR).
The interest rate can significantly affect how much money you earn on your savings or how much you pay on a loan:

• If you deposit money in a savings account, a higher interest rate means you will earn more on your principal over time.
• For loans, a higher interest rate means you will pay more interest over the life of the loan.

When calculating compound interest, the rate is expressed as \( r \), where \( r \) is in its decimal form. For example, a 3% interest rate is written as 0.03 in calculations. Therefore, understanding the impact of the interest rate helps in making informed financial decisions.

Principal Amount

The principal amount is the starting sum of money placed in an account or investment. It is the initial quantity of capital on which interest calculations are based. When you deposit money into a savings account or take a loan, the amount you initially deposit or borrow is called the principal.
Consider these key aspects of the principal:

• In terms of savings, the principal is the amount you start with, and its growth is driven by the interest rate.
• For loans, the principal is the actual amount borrowed, while interest payments are calculated on this sum.

In our compound interest scenario, from the exercise, the principal \( P \) is \( \$100 \). This is the fundamental amount that increases over time as the interest is applied each period. Knowing your principal is crucial because it forms the foundation on which all future interest is based, either growing your savings or determining your debt repayment.