Problem 78 - College Algebra

Question

Use the formula for the value of an annuity to solve,Round answers to the nearest dollar. To save money for a sabbatical to earn a master's degree, you deposit \(\$ 2500\) at the end of each year in an annuity that pays \(6.25 \%\) compounded annually. a. How much will you have saved at the end of five years? b. Find the interest.

Step-by-Step Solution

Answer

a. At the end of five years, $13,958 will be saved. b. The interest is $958.

Step 1: Identify the known values

The annual payment \(P = $2500\), the annual interest rate \(r = 6.25 \% = 0.0625\) and the number of periods \(n = 5\) years.

Step 2: Calculate the future value

Substitute the values into the annuity formula: \(F = P \times \frac{(1+r)^n - 1}{r}\) resulting in \(F = $2500 \times \frac{(1+0.0625)^5 - 1}{0.0625}\)

Step 3: Evaluate the expression

Calculate the expression gives \(F = $13,958\) (rounded to the nearest dollar)

Step 4: Calculate the interest

The interest is calculated by subtracting total payments from the future value: \(Interest = F - P \times n\) which is \(Interest = $13,958 - $2500 \times 5 = $958\)

Key Concepts

Compound Interest

Compound interest occurs when the interest earned on an investment is added back to the principal amount, so that in the next compounding period, interest is earned on both the original principal and the accrued interest from previous periods. Think of it as earning interest on interest, which allows the investment to grow at a faster rate compared to simple interest.

Key considerations include:

• Rate of interest: The percentage at which the compound interest accrues.
• Compounding frequency: How often the accumulated interest is added back to the principal. In the case of this problem, it's compounded annually.

At the end of the 5th year, both new contributions and accumulated interest are considered, making compound interest a powerful financial concept.

Future Value of Annuity

The future value of an annuity refers to the value of a series of cash flows at a specified date in the future. It's calculated by considering consistent payments (like annually depositing $2500) and the interest those payments earn over the investment period.

To calculate this, the formula used is:\[ F = P \times \frac{(1+r)^n - 1}{r} \]where:

• \( F \) is the future value of the annuity
• \( P \) is the annual payment
• \( r \) is the annual interest rate (converted into a decimal)
• \( n \) is the number of periods

This formula captures the accumulation of regular payments and their growth over time due to interest.

Annual Payment Calculation

An annual payment is the amount paid or received in regular installments over a year. In the context of an annuity, it is the consistent payment made at each period of the annuity's life. Here, the problem specifies an annual payment of $2500.

The annuity calculation assumes this payment is made at the end of each year.

• These payments grow according to the interest rate.
• The impact of each payment is determined by how much interest it can accumulate by the end of the period.

Knowing the annual payment allows us to assess the total contribution over time, which in this exercise amounts to $12,500 over five years.

Interest Calculation

Interest calculation involves determining the total amount of interest accumulated over the investment period. Here, it means calculating how much more money is earned or paid than originally deposited or borrowed.

The interest in this context can be found by subtracting the total amount of payments made from the future value of the annuity:\[ Interest = F - P \times n \]where:

• \( F \) is the future value calculated previously
• \( P \times n \) is the total contributions (in this case, \(12,500)

This results in the interest amount, which, after calculations (i.e., \)13,958 - \(12,500), yields \)958, the profit earned from investing in the annuity over five years.