Problem 58 - Precalculus

Question

If you put \(\$ 10,000\) in a savings account that earns \(3.5 \%\) interest per year compounded annually, how much would you expect to have in that account in 5 years?

Step-by-Step Solution

Answer

The account balance would be approximately $11,877 after 5 years.

Step 1: Identify the Formula

The formula to calculate the future value of an investment with compound interest is: \[ A = P (1 + \frac{r}{n})^{nt} \] where \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate (decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.

Step 2: Substitute Values into the Formula

Given in the problem: Principal \( P = 10,000 \), annual interest rate \( r = 0.035 \) (converted from 3.5%), \( n = 1 \) because the interest is compounded annually, and \( t = 5 \) years.Plug these values into the formula: \[ A = 10000 \times (1 + \frac{0.035}{1})^{1 \times 5} \]

Step 3: Calculate the Intermediate Values

First calculate \( \frac{0.035}{1} = 0.035 \).Then add 1 to get \( 1 + 0.035 = 1.035 \).Now raise this to the power of 5: \( 1.035^5 \).

Step 4: Compute the Final Amount

Calculate \( 1.035^5 \), which is approximately 1.1877.Multiply this by the principal: \[ A = 10000 \times 1.1877 = 11877 \]Therefore, the amount in the savings account after 5 years is approximately \ 11,877.

Key Concepts

Future Value of Investment

When you're looking at the future value of an investment, you want to figure out how much money you will have later on after interest has been added. This is particularly important for investments like savings accounts, where you earn interest over time. Using the formula for compound interest, you can predict this future value.

• The formula is: \( A = P (1 + \frac{r}{n})^{nt} \)
• A stands for the future value of the investment, or savings, including the interest earned.
• P is the principal (initial) amount of money put into the savings.

In our original exercise, by placing $10,000 in a savings account, we calculated how much that initial amount would grow in 5 years, assuming annual compounding. Think of this as the future payoff for setting your money aside for a given period.
Compounding is the process of earning interest on both the initial amount and the accumulated interest from previous periods. The key idea here is that the interest grows exponentially, which boosts your overall savings over time.

Annual Interest Rate

The annual interest rate is the percentage rate at which your investment grows each year. It's essentially a reward for depositing your money in a bank or an investment.

• This rate is usually expressed as a percentage. In our example, it was given as 3.5%.
• This number needs to be converted into a decimal when you use it in calculations. So, 3.5% becomes 0.035.
• An important thing to note is whether interest is compounded or simple. In our case, it's compounded annually, which means the interest is calculated on both the principal and the accumulated interest.

The more frequently interest is compounded, the more you will earn, because each compounding period adds to the total amount that's earning interest. If the interest were compounded quarterly or monthly, rather than annually, you'd see an even greater increase in the future value of your investment.

Investment Duration

Investment duration is simply the amount of time your money stays in the investment, to grow and earn interest. It answers the question, "How long will I keep this money invested?"

• It's a crucial factor in determining how much interest you will accumulate. The longer the money stays invested, the more interest you'll earn over time.
• In our step-by-step solution, the duration was 5 years.
• When using the compound interest formula, the duration influences the exponent \( nt \), since this is where the time factor is applied.
• Thus, for a given interest rate, the future value will be significantly higher if the investment duration is longer.

Investment duration plays a critical role, as compound interest tends to have a more pronounced effect over longer periods. So, the longer your money is invested, the greater the potential growth.