Business Calculus: Compound Interest and Exponential Growth Study Guide

Q1. Find the interest earned on $15,000 invested for 5 years at 7% interest compounded as follows: Annually, Semiannually, Quarterly, Monthly, and Continuously.

Background

Topic: Compound Interest and Continuous Compounding

This question tests your understanding of how to calculate interest earned using different compounding periods, including continuous compounding. This is a core application of exponential functions in business calculus.

Key Terms and Formulas

• Principal (P): The initial amount invested ($15,000).

• Annual Interest Rate (r): The yearly rate (7% or 0.07 as a decimal).

• Time (t): The number of years (5 years).

• Number of Compounding Periods per Year (n): Varies by compounding method.

Compound Interest Formula (for n times per year):

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Continuous Compounding Formula:

$A = Pe^{rt}$

Interest Earned:

$\text{Interest} = A - P$

Step-by-Step Guidance

1. Identify the values: $P = 15,000$, $r = 0.07$, $t = 5$ years.

2. For each compounding method, determine $n$:

   • Annually: $n = 1$

   • Semiannually: $n = 2$

   • Quarterly: $n = 4$

   • Monthly: $n = 12$

3. Plug the values into the compound interest formula for each case (except continuous):

   $A = 15,000 \left(1 + \frac{0.07}{n}\right)^{n \times 5}$

4. For continuous compounding, use:

   $A = 15,000 e^{0.07 \times 5}$

5. For each case, subtract the original principal to find the interest earned:

   $\text{Interest} = A - 15,000$

Try solving on your own before revealing the answer!

Q2. Assuming continuous compounding, what will it cost to buy a $50 item in 3 years at the following inflation rates: 3%, 4%, 5%?

Background

Topic: Exponential Growth and Continuous Compounding

This question asks you to apply the continuous compounding formula to model the effect of inflation on the price of an item over time.

Key Terms and Formulas

• Present Value (P): The current price ($50).

• Inflation Rate (r): The annual rate (as a decimal: 0.03, 0.04, 0.05).

• Time (t): Number of years (3).

Continuous Compounding Formula:

$A = Pe^{rt}$

Step-by-Step Guidance

1. Identify the values: $P = 50$, $t = 3$ years.

2. For each inflation rate, convert the percentage to a decimal (e.g., 3% = 0.03).

3. Plug the values into the formula for each rate:

   $A = 50 e^{r \times 3}$

4. Set up the calculation for each rate (but do not compute the final value yet).

Try solving on your own before revealing the answer!

Q3. Chris plans to invest $200 into a money market account. Find the interest rate needed for the money to grow to $1,800 in 14 years if the interest is compounded quarterly.

Background

Topic: Solving for Interest Rate in Compound Interest

This question tests your ability to solve for the interest rate in the compound interest formula, given the initial amount, final amount, time, and compounding frequency.

Key Terms and Formulas

• Principal (P): $200

• Final Amount (A): $1,800

• Time (t): 14 years

• Compounding Periods per Year (n): 4 (quarterly)

• Interest Rate (r): What you are solving for

Compound Interest Formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Step-by-Step Guidance

1. Write down the known values: $A = 1,800$, $P = 200$, $n = 4$, $t = 14$.

2. Set up the compound interest equation:

   $1,800 = 200 \left(1 + \frac{r}{4}\right)^{4 \times 14}$

3. Divide both sides by 200 to isolate the exponential expression:

   $\frac{1,800}{200} = \left(1 + \frac{r}{4}\right)^{56}$

4. Take the natural logarithm of both sides to solve for $r$ (but do not solve yet):

   $\ln\left(\frac{1,800}{200}\right) = 56 \ln\left(1 + \frac{r}{4}\right)$

Try solving on your own before revealing the answer!

Q4. Find the interest rate required for an investment of $5,000 to grow to $6,500 in 8 years if interest is compounded as follows (compounding periods not specified in the excerpt).

Background

Topic: Solving for Interest Rate in Compound Interest

This question is similar to Q3, but with different values. You may be asked to solve for $r$ for different compounding frequencies (e.g., annually, semiannually, etc.).

Key Terms and Formulas

• Principal (P): $5,000

• Final Amount (A): $6,500

• Time (t): 8 years

• Compounding Periods per Year (n): (Will depend on the specific part of the question)

• Interest Rate (r): What you are solving for

Compound Interest Formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Step-by-Step Guidance

1. Write down the known values: $A = 6,500$, $P = 5,000$, $t = 8$.

2. For each compounding method, set $n$ accordingly (e.g., $n = 1$ for annual, $n = 2$ for semiannual, etc.).

3. Set up the compound interest equation for each case:

   $6,500 = 5,000 \left(1 + \frac{r}{n}\right)^{n \times 8}$

4. Divide both sides by 5,000 to isolate the exponential expression:

   $\frac{6,500}{5,000} = \left(1 + \frac{r}{n}\right)^{8n}$

5. Take the natural logarithm of both sides to solve for $r$ (but do not solve yet):

   $\ln\left(\frac{6,500}{5,000}\right) = 8n \ln\left(1 + \frac{r}{n}\right)$

Try solving on your own before revealing the answer!