Problem 69 - Algebra and Trigonometry

Question

Compound Interest Find the time required for an investment of \(\$ 5000\) to grow to \(\$ 8000\) at an interest rate of 7.5\(\%\) per year, compounded quarterly.

Step-by-Step Solution

Answer

It takes approximately 7.64 years.

Step 1: Understand the Compound Interest Formula

The formula for compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (\$5000 here), \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time in years.

Step 2: Identify Known Values

We are given \( P = 5000 \), \( A = 8000 \), \( r = 0.075 \) (since 7.5% as a decimal is 0.075), and \( n = 4 \) (because the interest is compounded quarterly). We need to find \( t \).

Step 3: Substitute Values into the Formula

Plug the known values into the compound interest formula: \( 8000 = 5000 \left(1 + \frac{0.075}{4}\right)^{4t} \).

Step 4: Calculate the Interest Rate per Period

Calculate the interest rate per compounding period: \( \frac{0.075}{4} = 0.01875 \). Substitute this into the formula: \( 8000 = 5000 (1.01875)^{4t} \).

Step 5: Isolate the Exponential Expression

Divide both sides by 5000 to isolate the exponential expression: \( \frac{8000}{5000} = (1.01875)^{4t} \). Simplify the left side: \( 1.6 = (1.01875)^{4t} \).

Step 6: Apply Logarithms to Solve for 't'

Apply logarithms to both sides to solve for \( t \): \( \log(1.6) = 4t \cdot \log(1.01875) \).

Step 7: Simplify and Solve for 't'

Divide both sides by \( 4 \cdot \log(1.01875) \) to solve for \( t \): \( t = \frac{\log(1.6)}{4 \cdot \log(1.01875)} \).

Step 8: Calculate 't' Using Logarithms

Using a calculator, compute \( t \): \( t \approx \frac{0.2041}{4 \times 0.007989} \approx 7.64 \).

Key Concepts

Investment Growth

Investment growth refers to the increase in the value of an initial sum of money, or principal, over time due to interest earned on that principal. In our example, we begin with a principal, or initial investment, of $5000. We want this amount to grow to $8000.

• Growth occurs over periods due to interest accumulation.
• Compound interest plays a large role in how quickly this growth happens by adding earned interest back into the principal.

The idea is that the principal isn't the only thing earning interest; the interest itself also accumulates further interest over time. This compounding effect can significantly boost investment growth, especially over longer periods.

Exponential Functions

Exponential functions are mathematical functions used to model situations where a quantity grows or decays at a rate proportional to its current value. In the context of compound interest, the formula involves exponential functions to describe how the total account balance increases as time passes.
In our compound interest formula, \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), the term \( (1 + \frac{r}{n})^{nt} \) is the exponential part. It represents the repeated and exponentially compounding action of interest applied to the principal.

• The base of the exponential function here is \( (1 + \frac{r}{n}) \).
• The exponent is \( nt \), indicating how many times the interest is compounded over time, magnifying the growth effect.

Logarithms

Logarithms are the inverse operations of exponentiation and are very useful for solving equations where the exponent is unknown, as in our exercise. When we isolate the exponential expression in the compound interest calculation, we use logarithms to solve for the time variable, \( t \).
Consider the step where we reach the equation \( 1.6 = (1.01875)^{4t} \).

• We take the logarithm of both sides to bring down the exponent: \( \log(1.6) = 4t \cdot \log(1.01875) \).
• This manipulation allows us to solve for \( t \), the unknown time period required for the investment's growth from \(5000 to \)8000.

Logarithms thereby make it possible to solve for exponentiated variables in growth calculations.

Interest Rate Calculation

Interest rate calculation determines how much interest will apply to an investment over each compounding period. This calculation is pivotal in understanding how rapidly an investment will grow.
In our problem, the annual interest rate is 7.5%, but because the interest compounds quarterly, we have to adjust this rate to find the effective interest rate per period:

• First, convert the annual percentage into a decimal by dividing by 100: \( 0.075 \).
• Then, because the interest compounds four times a year, divide by 4: \( \frac{0.075}{4} = 0.01875 \).

Each quarter, the investment grows by roughly 1.875%, thanks to this periodic rate application. It's important to break down the interest rate according to how often it's compounded, as this influences the overall yield on an investment.