Problem 105 - Algebra and Trigonometry

Question

If \(\$ 2500\) is invested at \(3 \%\) compounded monthly, find the amount that results after a period of 2 years.

Step-by-Step Solution

Answer

The amount after 2 years is approximately $2654.59.

Step 1: Identify the formula

Use the compound interest formula: defined as \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]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 (in decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested for in years.

Step 2: Assign the values

Assign the given values to the variables in the formula:\( P = 2500 \)\( r = 0.03 \) (since 3% = 0.03)\( n = 12 \) (since compounded monthly)\( t = 2 \)

Step 3: Substitute values into the formula

Substitute the assigned values into the compound interest formula:\[ A = 2500 \left(1 + \frac{0.03}{12}\right)^{12 \times 2} \]

Step 4: Simplify the expression inside the parentheses

Calculate the value inside the parentheses:\[ 1 + \frac{0.03}{12} = 1 + 0.0025 = 1.0025 \]

Step 5: Compute the exponent

Calculate the exponent:\[ 12 \times 2 = 24 \]

Step 6: Evaluate the expression

Calculate the overall expression:\[ A = 2500 \left(1.0025\right)^{24} \]Use a calculator to find the value of \( (1.0025)^{24} \approx 1.061836546 \)Then, \[ A = 2500 \times 1.061836546 \approx 2654.59 \]

Key Concepts

investment growth

When you invest money, you expect it to grow over time. This process, where your initial amount (known as the principal) increases due to earning interest, is called investment growth. For example, if you invest \(\$ 2500\) at a \(3 \%\) annual interest rate compounded monthly, the total amount will increase each month. The longer you keep your investment, the more it will grow due to the compounding effect. To calculate this growth accurately, we use the compound interest formula.

interest rate

An interest rate is the percentage at which your investment grows over a period. For instance, in your exercise, the interest rate is \(3 \%\). Typically, interest rates can be compounded at different intervals, and understanding the rate's compounding frequency is crucial. A \(3 \%\) annual rate compounded monthly means each month, your investment grows by a fraction of the annual rate. More frequent compounding leads to faster growth, which we will easily calculate using the compound interest formula.

financial mathematics

Financial mathematics helps us understand and solve various financial problems using mathematical formulas. One key concept here is compound interest. By using the formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] you can predict how much your initial investment will grow over a certain period. Setting \(P = 2500\), \(r = 0.03\), \(n = 12\), and \(t = 2\) and substituting these values into the formula helps us calculate the final amount, illustrating how theoretical math concepts translate into real-world gains.

monthly compounding

Monthly compounding refers to the process where interest is calculated and added to your investment every month. This leads to the interest earning additional interest in subsequent months. It contrasts with annual compounding, where interest is added only once per year. Monthly compounding is more beneficial as your investment grows incrementally throughout the year. In our exercise, we use monthly compounding and observe how each small increment compounds multiple times over the two years, giving us a higher final amount.

exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of compound interest, the growth of investment follows an exponential function due to the repeated compounding of interest. The formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] shows exponential growth where \(n \times t\) acts as the exponent. For our specific case, the base \(1.0025\) is raised to the power of \(24\), showing the power of continuous growth. This exponential behavior illustrates the rapid accumulation of value over time.