Problem 13 - Algebra 1

Question

Find the balance after 5 years of an account that pays 4.8% interest compounded yearly given the following investment amounts. $$\$ 400$$

Step-by-Step Solution

Answer

The balance after 5 years of an account that started with $400 and pays 4.8% interest compounded yearly will be \$502.03.

Step 1: Understand the Problem

You need to find the balance after 5 years of an account that started with $400 and pays 4.8% interest compounded yearly.

Step 2: Convert the Percentage

Convert the annual interest rate from a percentage to a decimal by dividing by 100. So, the annual interest rate \(r = \frac{4.8}{100} = 0.048\).

Step 3: Use the Compound Interest Formula

Substitute the values for \(P = 400\), \(r = 0.048\), \(n = 1\) and \(t = 5\) into the formula \(A = P(1+\frac{r}{n})^{nt}\) to find \(A = 400(1+\frac{0.048}{1})^{1*5}\). Calculate the power first, then the multiplication inside the parenthesis, and finally the multiplication with the principal amount \(P = 400\). The balance after 5 years will be the result of this calculation.

Step 4: Do the Calculations

Calculating the values, we get \(A = 400(1+0.048)^{5} = 400*1.048^{5} = 400*1.25758 = \$502.03\). So, the balance after 5 years would be \$502.03

Key Concepts

Balance Calculation

When you invest money into an account that earns interest, you will want to know how much it will grow over time. The process of finding out this future balance is known as balance calculation. The key to calculating the future balance of an investment is understanding how compound interest works.

With compound interest, your initial investment, or principal, earns interest over time. Then, that interest starts earning interest itself. This is an example of a snowball effect, where your savings grow at an increasing rate.

To find the balance after a given period, you use the formula for compound interest:

• \(A = P(1 + \frac{r}{n})^{nt}\)

In this equation,

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the annual interest rate (in decimal).
• \(n\) is the number of times that interest is compounded per unit year.
• \(t\) is the time the money is invested for, in years.

The balance calculation shows how much an investment will be worth in the future, helping you plan effectively.

Interest Rate Conversion

Understanding interest rates is fundamental to managing investments. Interest rates are often provided as percentages, but they need to be converted into decimals when used in calculations.

Converting a percentage into a decimal is straightforward:

• Simply divide the percentage by 100. For example, a rate of 4.8% becomes \(0.048\).

This conversion is crucial for applying the compound interest formula, as it ensures the correct calculation of interest amounts.

Proper conversion ensures that when you use the interest rate in calculations, you achieve accurate results, reflecting the true growth of your investment.

Exponential Growth

Exponential growth is a concept where quantities grow at a consistent relative rate over a period. In the context of compound interest, this means the balance grows by a constant proportion each year.

The compound interest formula embodies exponential growth due to the variable \(n\) in the exponent \(nt\), which results in the balance being raised by a power. This power effect is what turns linear growth into exponential growth, leading to more pronounced increases over time.

This nature of growth makes compound interest a powerful tool for building wealth.

• It leverages the principle of 'interest on interest', allowing the balance to increase more significantly across longer time frames.
• The longer the investment period, the greater the compounding effect.

Understanding exponential growth helps individuals make informed decisions about saving and investing, aligning their strategies to maximize returns.