Question

For Exercises \(15-30,\) find the final balance and interest. \(\$ 2400\) at \(10 \%\) compounded quarterly for 4 years

Short answer

Based on the given information, calculate the future value of the investment and the interest earned after 4 years. Initial deposit: $2400 Annual interest rate: 10% Compounded quarterly Time: 4 years Future value: $3580.38 Interest earned: $1180.38

Step-by-step solution

Plug in the values into the formula

We have the formula \(A = P(1 + \frac{r}{n})^{nt}\). Plug in the values given in the exercise: \(A = 2400(1 + \frac{0.1}{4})^{4 \cdot 4}\)

Calculate the future value

First, calculate the value inside the parentheses: \(1 + \frac{0.1}{4} = 1 + 0.025 = 1.025\) Then, raise it to the power of \(nt = 4 \cdot 4 = 16\): \((1.025)^{16} = 1.4918246976\) (rounded to 10 decimal places) Now, multiply the initial deposit by the calculated value: \(A = 2400 \cdot 1.4918246976 = 3580.37927344\) Round the final balance to the nearest cent: \(A = \$3580.38\)

Calculate the interest earned

To calculate the interest earned, subtract the initial deposit (P) from the final balance (A): Interest = Final Balance - Initial Deposit Interest = \(3580.38 - 2400 = \$1180.38\) The final balance after 4 years is \(\$3580.38\) and the interest earned during this period is \(\$1180.38\).

Key concepts

Final Balance Calculation

Calculating the final balance in a compound interest scenario involves determining how much money you will have in your account after interest is applied over time. In this exercise, we start with a principal amount of \(2400 and an interest rate of 10%, compounded quarterly over 4 years. By using the compound interest formula, we aim to find the total amount or "balance" at the end of this period.

The formula used for the calculation is: \(A = P(1 + \frac{r}{n})^{nt}\), where:

• \(A\) is the final amount (balance)
• \(P\) is the principal amount (initial deposit)
• \(r\) is the annual interest rate
• \(n\) is the number of times the interest is compounded per year
• \(t\) is the time the money is invested for, in years

Using this, we find that the final balance in this specific exercise is \\)3580.38 after rounding.

Interest Calculation

Interest calculation in compound interest problems helps us understand how much money has been earned or accumulated over time, separate from the initial principal. To find the interest earned, we subtract the initial deposit from the final balance.

In our exercise, we calculated the final balance as \\(3580.38. The interest earned can thus be determined by the formula:

Interest Earned = Final Balance - Initial Deposit

So, \(Interest = 3580.38 - 2400 = \\)1180.38\)

This amount represents the profit made from the investment due to the effects of compound interest, showing how your money grows over time.

Quarterly Compounding

Quarterly compounding is a method where the interest is calculated and added to the principal four times a year. This means every three months, the earned interest is added to the original amount, and the new total becomes the principal for the next period.

In our example, the 10% interest rate is divided across the four quarters of each year, resulting in a rate of \(\frac{0.1}{4} = 0.025\) or 2.5% per quarter.

This frequent compounding leads to a faster growth of the investment because interest is applied more often, compared to annual or even semi-annual compounding. Understanding the frequency of compounding helps in comprehending how your investments accumulate more returns over time.

Future Value Formula

The future value formula is essential in determining how much a current investment will grow over a certain period, considering compound interest. It expresses the potential growth of an investment by factoring in the time, interest rate, and the frequency of compounding.

Our main formula, \(A = P(1 + \frac{r}{n})^{nt}\), plays a pivotal role in this calculation. With every element representing a crucial part of the computation:

• \(A\) is the future value after interest has been compounded.
• \(P\) represents the start amount or principal.
• \(r\) is the interest rate per annum.
• \(n\) shows how many times the interest is compounded annually.
• \(t\) describes the duration of the investment in years.

Using this formula helps one predict how much an initial investment could be worth in the future, providing a financially insightful view into the power of compounding.