Question

The compound interest formula is \(A=P(1+i)^{n},\) where \(A\) is the future amount, \(P\) is the present amount or principal, \(i\) is the interest rate per compounding period expressed as a decimal, and \(n\) is the number of compounding periods. All interest rates are annual percentage rates (APR). a) David inherits \(\$ 10\) 000 and invests in a guaranteed investment certificate (GIC) that earns \(6 \%,\) compounded semi-annually. How long will it take for the GIC to be worth \$11 000? b) Linda used a credit card to purchase a S1200 laptop computer. The rate of interest charged on the overdue balance is \(28 \%\) per year, compounded daily. How many days is Linda's payment overdue if the amount shown on her credit card statement is \(\$ 1241.18 ?\) c) How long will it take for money invested at \(5.5 \%,\) compounded semi- annually, to triple in value?

Short answer

a) 1.61 years. b) 45 days. c) 20.13 years.

Step-by-step solution

Title - Identify Variables for Part (a)

For the GIC investment problem, identify the given variables: \(P = 10,000\), \(A = 11,000\), \(i = \frac{0.06}{2} = 0.03\) (since 6% is annually and we need semi-annual rate), and solve for \(n\).

- Use the Compound Interest Formula for Part (a)

Substitute the identified values into the formula: \[ 11,000 = 10,000(1 + 0.03)^n \].

- Solve for Part (a)

Divide both sides by 10,000: \[ 1.1 = (1.03)^n \]. Take the natural logarithm of both sides: \[ \ln(1.1) = n \ln(1.03) \]. Solve for \(n\): \[ n = \frac{\ln(1.1)}{\ln(1.03)} \approx 3.22 \]. Hence, \(n\approx 3.22\) semi-annual periods or about 1.61 years.

Title - Identify Variables for Part (b)

For the credit card problem, identify the variables: \(P = 1200\), \(A = 1241.18\), \(i = \frac{0.28}{365} \approx 0.000767\) (since it’s compounded daily), and solve for \(n\).

- Use the Compound Interest Formula for Part (b)

Substitute the given values into the formula: \[ 1241.18 = 1200 (1 + 0.000767)^n \].

- Solve for Part (b)

Divide both sides by 1200: \[ \frac{1241.18}{1200} \approx 1.03432 = (1.000767)^n \]. Take the natural logarithm of both sides: \[ \ln(1.03432) = n \ln(1.000767) \]. Solve for \(n\): \[ n = \frac{\ln(1.03432)}{\ln(1.000767)} \approx 44.92 \]. Hence, \(n\approx 45\) days.

Title - Identify Variables for Part (c)

For the investment to triple problem, identify the variables: \(P = P\), \(A = 3P\), \(i = \frac{0.055}{2} = 0.0275\) (since 5.5% APR compounded semi-annually), and solve for \(n\).

- Use the Compound Interest Formula for Part (c)

Substitute the given values into the formula: \[ 3P = P(1 + 0.0275)^n \].

- Solve for Part (c)

Divide both sides by \(P\): \[ 3 = (1.0275)^n \]. Take the natural logarithm of both sides: \[ \ln(3) = n \ln(1.0275) \]. Solve for \(n\): \[ n = \frac{\ln(3)}{\ln(1.0275)} \approx 40.25 \]. Hence, \(n\approx 40.25\) semi-annual periods or about 20.13 years.

Key concepts

Future Amount Calculation

The future amount, often abbreviated as **A**, is the total amount of money you will have after investing or saving for a certain period, including the interest earned. This concept is crucial when understanding compound interest, as it helps you calculate how much your investment will grow over time.

To find the future amount, you need the compound interest formula: \( A = P(1 + i)^n \). Here's what each term means:
- **A**: Future amount
- **P**: Principal (initial amount)
- **i**: Interest rate per period
- **n**: Number of compounding periods.

For instance, if you invest \(10,000\) dollars at an annual interest rate of 6%, compounded semi-annually, to find the future amount after, say, 3 years (which is 6 periods), you would substitute these values into the formula.

Interest Rate Per Period

When dealing with the compound interest formula, it's important to use the correct interest rate for each compounding period, not just the annual rate.

If the interest is compounded semi-annually (twice a year), you divide the annual interest rate by 2. For example, a 6% annual interest rate becomes \(\frac{0.06}{2} = 0.03\) per period. Similarly, for daily compounding, you would divide the annual rate by the number of days in a year, typically 365. So, 28% annual interest, compounded daily, becomes \(\frac{0.28}{365} \approx 0.000767 \).

If you are unsure, always double-check the compounding frequency and make sure to adjust the interest rate accordingly.

Compounding Periods

The number of compounding periods, denoted as **n**, significantly impacts how much interest accrues on an investment or debt.

Compounding can occur on different schedules:

• Daily
• Monthly
• Quarterly
• Semi-Annually
• Annually

Each of these schedules affects how often interest is calculated and added to the principal. For example, semi-annual compounding means interest is compounded twice a year. Thus, 10 years of semi-annual compounding results in \(10 \times 2 = 20\) periods.

The more frequently interest is compounded, the more you earn on your investment. This is why understanding the number of compounding periods is essential.

Natural Logarithm

The natural logarithm, commonly denoted as **ln**, plays a crucial role in solving compound interest problems, especially when you need to find the number of compounding periods, **n**.

In the formula \( A = P(1 + i)^n \), if you need to solve for **n**, you might not be able to do so through basic algebra alone. Taking the natural logarithm of both sides of the equation helps simplify it:

\( A = P(1 + i)^n \) becomes \( ln(A) = ln(P(1 + i)^n) \).

Since \( ln(a^b) = b \times ln(a) \), the equation simplifies to \( ln(A) = n \times ln(1 + i) \).

Solve for **n** by isolating it: \( n = \frac{ln(A)}{ln(1 + i)} \).

This shows how the natural logarithm is indispensable for unravelling the time period required for an investment to reach a specific future amount.