Precalculus Financial Models: Compound Interest, Doubling/Tripling Time, and Interest Rate Problems

Q1. What rate of interest compounded annually is needed in order to double an investment in 7 years?

Background

Topic: Compound Interest and Exponential Equations

This question tests your ability to use the compound interest formula to solve for the interest rate required to double an investment over a specified period of time. This is a classic application of exponential growth in financial mathematics.

Key Terms and Formulas

• Compound Interest Formula:

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

• $A$ = final amount

• $P$ = principal (initial amount)

• $r$ = annual interest rate (as a decimal)

• $n$ = number of compounding periods per year

• $t$ = number of years

For annual compounding, $n = 1$.

Step-by-Step Guidance

1. Let $P$ be the initial investment. To double the investment, set $A = 2P$.

2. Substitute $A = 2P$, $n = 1$, and $t = 7$ into the compound interest formula:

   $2P = P(1 + r)^7$

3. Divide both sides by $P$ to simplify:

   $2 = (1 + r)^7$

4. Take the 7th root of both sides to solve for $r$:

   $1 + r = 2^{1/7}$

5. Isolate $r$ by subtracting 1 from both sides:

   $r = 2^{1/7} - 1$

Try solving on your own before revealing the answer!

Final Answer: $r \approx 0.1041$ or 10.41%

We used the formula $r = 2^{1/7} - 1$ to find the annual interest rate needed to double the investment in 7 years.

Q2. a) How long will it take for an investment to double in value if it earns 6% compounded quarterly?

Background

Topic: Compound Interest and Solving for Time

This question asks you to determine the time required for an investment to double, given a specific interest rate and compounding frequency. This involves solving an exponential equation for the variable $t$.

Key Terms and Formulas

• Compound Interest Formula:

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

• $A$ = final amount

• $P$ = principal (initial amount)

• $r$ = annual interest rate (as a decimal)

• $n$ = number of compounding periods per year (quarterly: $n = 4$)

• $t$ = number of years

Step-by-Step Guidance

1. Let $P$ be the initial investment. To double the investment, set $A = 2P$.

2. Substitute $A = 2P$, $r = 0.06$, $n = 4$ into the formula:

   $2P = P \left(1 + \frac{0.06}{4}\right)^{4t}$

3. Divide both sides by $P$ to simplify:

   $2 = (1.015)^{4t}$

4. Take the natural logarithm of both sides to bring down the exponent:

   $\ln 2 = 4t \ln 1.015$

5. Solve for $t$:

   $t = \frac{\ln 2}{4 \ln 1.015}$

Try solving on your own before revealing the answer!

Final Answer: $t \approx 11.64$ years

We used logarithms to solve for $t$ in the compound interest equation with quarterly compounding.

Q3. b) How long will it take for an investment to double in value if it earns 6% compounded continuously?

Background

Topic: Continuous Compounding and Solving for Time

This question tests your understanding of the continuous compounding formula and your ability to solve for the time variable using logarithms.

Key Terms and Formulas

• Continuous Compounding Formula:

$A = Pe^{rt}$

• $A$ = final amount

• $P$ = principal (initial amount)

• $r$ = annual interest rate (as a decimal)

• $t$ = number of years

Step-by-Step Guidance

1. Let $P$ be the initial investment. To double the investment, set $A = 2P$.

2. Substitute $A = 2P$ and $r = 0.06$ into the formula:

   $2P = Pe^{0.06t}$

3. Divide both sides by $P$ to simplify:

   $2 = e^{0.06t}$

4. Take the natural logarithm of both sides:

   $\ln 2 = 0.06t$

5. Solve for $t$:

   $t = \frac{\ln 2}{0.06}$

Try solving on your own before revealing the answer!

Final Answer: $t \approx 11.55$ years

We used the continuous compounding formula and logarithms to solve for the doubling time.

Q4. c) How long will it take for an investment to triple in value if it earns 6% compounded continuously?

Background

Topic: Continuous Compounding and Solving for Time

This question extends the previous one by asking for the time required to triple an investment, using the continuous compounding formula.

Key Terms and Formulas

• Continuous Compounding Formula:

$A = Pe^{rt}$

• $A$ = final amount

• $P$ = principal (initial amount)

• $r$ = annual interest rate (as a decimal)

• $t$ = number of years

Step-by-Step Guidance

1. Let $P$ be the initial investment. To triple the investment, set $A = 3P$.

2. Substitute $A = 3P$ and $r = 0.06$ into the formula:

   $3P = Pe^{0.06t}$

3. Divide both sides by $P$ to simplify:

   $3 = e^{0.06t}$

4. Take the natural logarithm of both sides:

   $\ln 3 = 0.06t$

5. Solve for $t$:

   $t = \frac{\ln 3}{0.06}$

Try solving on your own before revealing the answer!

Final Answer: $t \approx 18.3$ years

We used the continuous compounding formula and logarithms to solve for the tripling time.