Problem 22 - Algebra 1: Concepts and Skills

Question

You deposit money in an account that pays 5% interest compounded yearly. Find the balance after 5 years for the given initial amount. $$\$ 350$$

Step-by-Step Solution

Answer

The balance in the account after 5 years will be approximately $444.16.

Step 1: Convert the percentages into decimals

First, change the interest rate from a percentage into a decimal format by dividing by 100. In this case, r = 5% = 0.05.

Step 2: Substitute the values into the formula

Next, substitute the values into the formula A = P(1 + r/n)^(nt). This gives: A = $350(1 + 0.05/1)^(1*5).

Step 3: Simplify Mathematical Expression

Now, simplify the expression inside the parentheses and then raise it to the power as indicated by the formula. After this, multiply it by the principal amount. This gives: A = $350 * (1.05)^5.

Step 4: Perform the calculation

Finally, perform the calculation. The final amount in the account after five years will be approximately $444.16. This sum represents the initial deposit plus the compounded interest over the five year period.

Key Concepts

Understanding Interest Rate

Interest rates are the percentages that help determine how much money your account will grow over time. They indicate what fraction of the initial deposit, or principal, is added to your account balance annually.
When the bank or financial institution gives a 5% interest rate, it implies that 5% of the initial amount you deposited will be added to your account balance after each year.

• A higher interest rate means more growth of your initial deposit.
• A lower interest rate means slower growth over the same time period.

Understanding the interest rate is crucial for predicting how much your initial deposit will increase over time.

Defining Initial Deposit

The initial deposit is the original amount of money you place into your account. It serves as the starting point for calculating growth through interest. In our example, this amount is $350.
Every year, the interest rate is applied to this initial deposit to calculate how much the account will grow.

• A larger initial deposit results in a bigger balance over time when compounded with interest.
• The initial deposit does not change during the compounding period, but it directly affects the amount of interest gained.

Understanding your initial deposit helps you see the potential of your financial growth through compounded interest.

Yearly Compounding Explained

Yearly compounding refers to the interest calculation and addition to the account balance occurring once per year. This is a common compounding frequency and is straightforward to understand.
In the formula used in our example, yearly compounding is represented by setting the compounding frequency, n, to 1.
This means that, after each year, the interest is calculated on the current balance including any interest that was added from previous years.

• With yearly compounding, once a year, the interest earned is added to your balance.
• Each year, the interest compounds on both the initial deposit and the accumulated interest from prior years.

Yearly compounding helps your savings grow at a steady pace because each year's interest is calculated on a new, higher balance.

Balance Calculation Process

To find the future balance of an account with compounded interest, you use a specific formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

• "\(A\)" represents the final amount or balance.
• "\(P\)" is the initial deposit or principal amount.
• "\(r\)" stands for the annual interest rate in decimal form.
• "\(n\)" is the number of compounding periods per year.
• "\(t\)" is the time the money is invested for, measured in years.

For the example provided:- Substitute \(P = 350\), \(r = 0.05\), \(n = 1\), and \(t = 5\).- The calculation, \(A = 350 \times (1 + 0.05)^5\), simplifies to \(A = 350 \times 1.27628156\).- The final account balance after 5 years is approximately $444.16.This calculation shows how our money grows each year when compounded annually, integrating both the initial deposit and the interest accumulated over this period.