Problem 34 - College Algebra

Question

31 to calculate the initial d… # For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) Use the formula found in Exercise #31 to calculate the initial deposit of an account that is worth $14,472.74 after earning 5.5% interest compounded monthly for 5 years. (Round to the nearest dollar.)

Step-by-Step Solution

Answer

The initial deposit is $10,970.

Step 1: Understand the Compound Interest Formula

The compound interest formula is given by \( A(t) = P \left(1 + \frac{r}{n} \right)^{nt} \), where \( A(t) \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate (in decimal), \( n \) is the number of times interest is compounded per year, and \( t \) is the time in years.

Step 2: Identify Given Values

We are given: \( A(t) = 14,472.74 \), \( r = 5.5\% = 0.055 \), \( n = 12 \) (compounded monthly), and \( t = 5 \) years. We need to find \( P \), the initial deposit.

Step 3: Rearrange Formula to Solve for P

We need to rearrange the formula to find \( P \). This is done by dividing both sides by \( \left(1 + \frac{r}{n} \right)^{nt} \), resulting in:\[ P = \frac{A(t)}{ \left(1 + \frac{r}{n} \right)^{nt} } \]

Step 4: Calculate the Denominator

Calculate the value of the denominator \( \left(1 + \frac{r}{n} \right)^{nt} \). Substitute the values: \( 1 + \frac{0.055}{12} = 1.0045833 \), and then raise this to the power of \((12 \times 5) = 60\):\[ (1.0045833)^{60} \approx 1.31907 \]

Step 5: Calculate Initial Deposit P

Substitute \( A(t) = 14,472.74 \) and the calculated denominator into the rearranged formula:\[ P = \frac{14,472.74}{1.31907} \approx 10,970 \]Round to the nearest dollar to obtain \( P = 10,970 \).

Step 6: Answer Check

Verify if the calculations appear correct. Recalculating with the given values confirms that the initial deposit for the account to be worth $14,472.74 after 5 years, with a 5.5% interest rate compounded monthly, is approximately $10,970.

Key Concepts

Initial Deposit Calculation

To find the initial deposit required in a compound interest scenario, we use the compound interest formula, which helps determine how an investment grows over time. The formula is \( A(t) = P \left(1 + \frac{r}{n} \right)^{nt} \), where:

• \( A(t) \) is the future value of the investment/loan, including interest.
• \( P \) is the principal amount (initial deposit).
• \( r \) is the annual interest rate, expressed as a decimal.
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for, in years.

In this exercise, we reverse the formula to solve for \( P \), resulting in \( P = \frac{A(t)}{ \left(1 + \frac{r}{n} \right)^{nt} } \). This allows us to calculate the initial deposit needed to reach a future value given a specific interest rate and compounding frequency.

Compounded Monthly

Compounding interest means that the interest earned over time is added to the original amount, so future interest is calculated on both the initial principal and the accumulated interest.
When interest is compounded monthly, it implies that the interest is calculated and added to the account balance 12 times a year.

• This increases the total amount of interest earned compared to less frequent compounding periods, like annually.
• In our formula, \( n = 12 \), representing monthly compounding.

Monthly compounding can significantly increase an investment's value over time, making it a powerful tool in growing savings or investments.

Annual Interest Rate

The annual interest rate percent, represented by \( r \) in the compound interest formula, is typically provided as a percentage, but it must be converted to a decimal form when used in formulas. For instance, a rate of 5.5% converts to \( r = 0.055 \).
This rate indicates how much interest will accumulate over a year excluding the effects of compounding.

• Higher rates naturally lead to more significant returns on investments.
• Even small changes in the interest rate can lead to significant differences in final amounts due to compounding effects.

It's essential to understand how the nominal annual rate will behave once subjected to compounding to correctly predict investment growth.

Time in Years

The time variable \( t \) in the compound interest formula indicates the total duration for which the money is invested or borrowed, measured in years.
This period is fundamental in understanding how much an investment can grow.

• In the given problem, \( t = 5 \) years, meaning interest compounds over five years.
• The longer the time frame, the more interest will be compounded, leading to exponential growth due to the repeated application of interest on the increasing account balance.

Time works together with the compounding frequency and rate to determine the future value of an investment or debt, emphasizing the benefits of long-term investing.