Problem 4 - Algebra 1: Concepts and Skills

Question

You deposit \(\$500\) in an account that pays 4% interest compounded yearly. Complete this equation to write an exponential growth model for the balance after t years: \(A=?(1+?)^{t}\)

Step-by-Step Solution

Answer

Therefore, the equation that models the balance in the account after t years is \(A = 500(1.04)^{t}\).

Step 1: Identify the principal amount

The principal amount P is the initial amount of money that is put into the account. From the problem, it is given that $500 is deposited. So, P = 500.

Step 2: Identify the annual interest rate

The annual interest rate r is the percentage of the principal amount that is added on annually. In this problem, the annual interest rate is given as 4%. When using the interest rate in the formula, it needs to be converted from percentage to decimal form. So, r = 4/100 = 0.04.

Step 3: Write the formula

Now using the values of P and r, the formula becomes \(A = 500(1+0.04)^{t}\), or simply \(A = 500(1.04)^{t}\). This is the formula for the account balance after t years.

Key Concepts

Compound Interest

Compound interest is a powerful concept in finance that allows your investments to grow over time, not just on your initial deposit or principal, but also on the interest that accumulates. Unlike simple interest, which is calculated only on the principal amount, compound interest includes the interest on both the initial amount and the accumulate interest from previous periods.

Here's how it works: when you deposit money into an account that compounds interest, the interest is added to the principal at each compounding period. In our example, this occurs annually. The next year's interest is then calculated on the new principal, which now includes the interest from the previous year.

• The formula for compound interest is: \( A = P(1 + r)^t \), where \( A \) is the amount after time \( t \), \( P \) is the principal, \( r \) is the annual interest rate, and \( t \) is the number of years.
• This formula shows how compound interest results in exponential growth of your investment over time.

Exponential Functions

Exponential functions are mathematical expressions that describe situations where quantities grow or decay at a constant relative rate. They are represented in the form \( y = a(1 + r)^t \), and relevant in multiple fields such as finance, science, and population studies. In this context, we'll focus on their application in finance through compound interest.

When you deposit money in a bank account with compound interest, the balance grows exponentially. This means that, rather than growing by the same amount each year, the balance grows by the same percentage each year. As a result, the increase in the balance becomes larger over time.

• In our exponential growth model, \( A = 500(1.04)^t \), the base \( 1.04 \) represents the annual growth factor, composed of the principal untouched plus the interest rate expressed as a decimal.
• This model underscores how powerful exponential growth can be, especially as the number of years \( t \) increases.

By understanding exponential functions, you can better grasp how your investments can increase significantly over long periods due to compounding.

Interest Rate

The interest rate is a crucial factor in determining how much you'll earn on your savings or investments over time. An interest rate is essentially the percentage at which your money will grow in an account for each compounding period.

In our example, the annual interest rate is 4%. This rate determines the growth of the account balance every year. When applying the interest rate in formulas, it's essential to convert the percentage to its decimal form for calculations. For instance, 4% becomes 0.04.

• With compound interest, the effect of even small changes in the interest rate can accumulate into significant differences over time.
• Higher interest rates can greatly accelerate the rate of growth of your investment due to the compounding effect.

Understanding the impact of the interest rate can help in making wiser financial decisions and choosing the best savings or investment options available.