Problem 5 - Prealgebra

Question

Savings Account A local bank pays \(5.5 \%\) annual interest on all savings accounts. If \(\$ 600\) is invested in this type of account, how much will be in the account at the end of a year? Bank Loan A farmer borrows \(\$ 8,000\) from his local bank at \(7 \% .\) How much does he pay back to the bank at the end of the year when he pays off the loan?

Step-by-Step Solution

Answer

Total in savings: \$633; Total loan repayment: \$8,560.

Step 1: Understand Simple Interest for Savings

For a savings account with no additional deposits or withdrawals, we apply the concept of simple interest. Simple interest is calculated using the formula \( I = P \times r \times t \), where \( I \) is the interest, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years. Here, \( P = \$600 \), \( r = 5.5\% = 0.055 \), and \( t = 1 \).

Step 2: Calculate Interest for Savings

Using the simple interest formula: \( I = 600 \times 0.055 \times 1 \). Calculate the interest earned over one year.

Step 3: Add Interest to Principal for Savings

The total amount in the savings account after one year is the sum of the principal and the interest. Compute \( A = P + I = 600 + \text{Interest from Step 2} \).

Step 4: Understand Simple Interest for Loans

For the bank loan, the farmer must repay the amount borrowed plus interest. Like before, use the simple interest formula \( I = P \times r \times t \). Here, \( P = \$8000 \), \( r = 7\% = 0.07 \), and \( t = 1 \).

Step 5: Calculate Interest for Loan

Use the formula \( I = 8000 \times 0.07 \times 1 \) to find the interest on the loan for the year.

Step 6: Calculate Total Repayment for Loan

The total repayment amount is the sum of the principal and the interest. Compute \( A = P + I = 8000 + \text{Interest from Step 5} \).

Key Concepts

Understanding Savings Accounts

Savings accounts are financial tools offered by banks where you can deposit money and earn interest over time. They are popular for storing funds securely while ensuring that your money grows, albeit at a modest rate.

Here’s why savings accounts matter:

• Security: Money in savings accounts is generally insured by the government up to a certain amount, making it a safe place to store funds.
• Convenience: These accounts often come with easy access features like online banking and ATM access.
• Interest Earnings: Banks pay you interest in exchange for holding your money, calculated on the balance over a period.

Understanding how savings account interest works is crucial. Simple interest is a common formula used to calculate the earnings from savings accounts. It uses the formula\[ I = P \times r \times t \]where \( I \) denotes the interest earned, \( P \) is your principal deposit, \( r \) represents the annual interest rate, and \( t \) is the duration in years.

For example, depositing \$600 in a savings account with a \(5.5\%\) annual interest rate will earn you interest without further deposits or withdrawals. At the end of a year, the bank will have added this interest to your principal, thus growing your total balance.

Decoding Bank Loans

Bank loans are sums of money borrowed from financial institutions that must be repaid with interest. They are vital for funding personal or business needs, such as buying a home, paying for education, or even purchasing a car.

Key points about bank loans:

• Principal Amount: This is the original sum borrowed that you'll need to repay.
• Interest: This is an additional amount calculated on the principal that you’ll pay back, which is how banks earn from giving loans.

When you're borrowing money, the same simple interest formula can apply. For a loan taken out, it’s calculated by\[ I = P \times r \times t \]where \( I \) represents the interest to be paid over the term, \( P \) is the principal amount borrowed, \( r \) is the annual interest rate, and \( t \) is the loan term in years.

For instance, a farmer borrowing \$8000 at a \(7\%\) rate will compute the interest for one year, adding it to the principal to find out the total repayment at the end of that year, making it clear how loans grow in cost over time.

How the Annual Interest Rate Works

The annual interest rate is a percentage that represents the cost of borrowing money or the return on savings for one year. This rate is central to calculating how much interest you will pay or earn.

Important points to know:

• Percentage Form: It's expressed as a percentage (e.g., \(5.5\%\) for savings, \(7\%\) for loans).
• Time Factor: Always applies over the course of one year, impacting both loans and savings calculations.
• Simplicity: Applying this rate through the simple interest formula helps in straightforward computations.

When dealing with finances, the annual interest rate gives you a clear view of how your money grows in savings or how your debt increases with loans. By using\[ I = P \times r \times t \]the simplicity is evident. For savings, it indicates how much extra the bank pays for the use of your money, for loans, it shows the additional cost for borrowing.

Understanding the annual interest rate is crucial for making informed financial decisions, whether you're saving for a future need or managing loan repayments effectively.