Question

Your savings account currently has a balance of \(\$ 32,300\). You opened the savings account two years ago and have not added to the initial amount you deposited. If your savings have been earning an annual interest rate of 2 percent, compounded annually, what was the amount of your original deposit?

Short answer

The original deposit was approximately $31,050.

Step-by-step solution

Understand the Problem

We're given the current balance of a savings account, the annual interest rate, and the number of years the money has been in the account. We need to determine the original deposit amount.

Use the Compound Interest Formula

The formula for compound interest when interest is compounded annually is: \[ A = P (1 + r)^t \]Where \(A\) is the amount of money accumulated after \(t\) years, including interest, \(P\) is the principal amount (initial deposit), \(r\) is the annual interest rate, and \(t\) is the number of years the money is invested or borrowed.

Substitute the Known Values

In the formula \( A = P (1 + r)^t \), we substitute \(A = 32,300\), \(r = 0.02\), and \(t = 2\) years.\[ 32,300 = P (1 + 0.02)^2 \]

Simplify the Equation

Calculate the term \((1 + 0.02)^2\):\[ (1.02)^2 = 1.0404 \]So the equation becomes:\[ 32,300 = P \times 1.0404 \]

Solve for the Principal Amount

Rearrange the equation to solve for \(P\): \[ P = \frac{32,300}{1.0404} \]Now, divide to find \(P\):\[ P \approx 31,050 \]

Verify Your Result

Check the calculation by plugging \(P\) back into the compound interest formula to ensure it results in the current account balance. If \(31,050\) is multiplied by \(1.0404\), the result should be close to \(32,300\), confirming our computation is correct.

Key concepts

Savings Account

A savings account is a secure place to store your money while earning interest over time. When you open a savings account, your intention is likely to keep your funds safe while growing them gradually. The financial institution that holds your account pays you interest, which is a small percentage of your balance, as a reward for keeping your money with them.
When funds are deposited into a savings account, they are often referred to as the "principal". This is the initial amount you deposit. Over time, due to the effects of compound interest, this amount will grow.

• A savings account is ideal for short-term financial goals, such as saving for a vacation or creating an emergency fund.
• It provides liquidity, meaning you can access your money relatively easily when you need it.
• Savings accounts are usually insured by government bodies, adding an extra layer of security.

Understanding how a savings account works is a crucial part of effective financial planning.

Interest Rate

The interest rate is the percentage of your savings that the bank pays you for keeping your money with them. It's an essential aspect of how much your money will grow over time. In our example, the savings account has an annual interest rate of 2%.
The interest on savings accounts is typically compounded annually, which means interest is added to your principal at the end of each year. The next year's interest is calculated on this new amount.

• An interest rate of 2% means you'll earn 2% of your account balance as interest each year.
• The higher the interest rate, the faster your savings grow.
• Interest rates can vary widely between financial institutions and account types.

Choosing a savings account with a favorable interest rate will help maximize your earnings over time.

Original Deposit

The original deposit, also known as the principal, is the initial amount of money you place into your savings account. In our example, we're tasked to find out what the original deposit was two years ago.
We know the balance now is \(32,300, and it has been growing at a 2% annual interest rate due to compounding. To find the original deposit, we use the compound interest formula: \[ A = P (1 + r)^t \]Where:- \(A\) is the savings account balance now (\)32,300)- \(r\) is the annual interest rate (0.02)- \(t\) is the time in years (2).
This formula allows us to work backward to find \(P\), the original deposit.
Solving the equation: \[ 32,300 = P (1.02)^2 \] leads us to find: \[ P \approx 31,050 \]This calculation shows that about $31,050 was the initial amount deposited.

Financial Literacy

Financial literacy refers to the understanding of how money works and making informed decisions in personal finances. It's an essential skill in today’s society for managing savings, investments, and expenses effectively. Being financially literate means recognizing how savings accounts, interest rates, and compound interest interact to grow your wealth.
Educating yourself about financial concepts can help you:

• Make better decisions about where to save and invest your money.
• Understand the benefits and pitfalls of different financial products.
• Create a solid financial plan that includes savings and investments.

Financial literacy empowers individuals to make choices that lead to financial well-being and prosperity throughout their lives.