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Magazine Chapter 1: Problem 30
Investment Suppose that an investment of \(\$ 1000\) increases by \(8 \%\) per year. a. Write a formula for the value of this investment after \(t\) years. b. How long will it take this investment to double?
Short Answer
Expert verified
The formula is \( A = 1000(1.08)^t \). It takes approximately 9 years to double.
Step by step solution
01
Understanding Compound Interest Formula
To find the problem's solution, first understand that the investment increases by a fixed percentage each year. This suggests compound interest. The general formula for compound interest is given by \( A = P(1 + r)^t \) where \( A \) is the amount of money accumulated after \( t \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (in decimal), and \( t \) is the time the money is invested for in years.
02
Setting Up the Formula for Future Value
Here, the principal \( P \) is \( 1000 \) and the rate \( r \) is \( 0.08 \) (8% converted to decimal). Substitute these values into the compound interest formula: \( A = 1000(1 + 0.08)^t \). Simplify it to: \( A = 1000(1.08)^t \). This formula calculates the investment's value after \( t \) years.
03
Solving for Doubling Time
To find out how long it takes for the investment to double, set \( A = 2000 \) (since doubling \( 1000 \) results in \( 2000 \)). Substitute into the formula: \( 2000 = 1000(1.08)^t \). Simplify to \( 2 = (1.08)^t \), because \( \frac{2000}{1000} = 2 \).
04
Solving the Exponential Equation
To solve \( 2 = (1.08)^t \), take the natural logarithm of both sides to eliminate the exponent: \( \ln(2) = \ln((1.08)^t) \). By the properties of logarithms, \( \ln((1.08)^t) = t \cdot \ln(1.08) \). Thus, \( \ln(2) = t \cdot \ln(1.08) \).
05
Calculating Value of t
Isolate \( t \) by dividing both sides by \( \ln(1.08) \): \( t = \frac{\ln(2)}{\ln(1.08)} \). Compute using a calculator: \( \ln(2) \approx 0.6931 \) and \( \ln(1.08) \approx 0.07696 \). Thus, \( t \approx \frac{0.6931}{0.07696} \approx 9.006 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Investment Growth
Investment growth refers to the increase in the value of an investment over time. When you invest your money, the primary goal is to make it grow.
Different factors can influence this growth, but one of the most common is the rate of return or the interest rate. In our exercise, we have an investment of \( \$1000 \) growing at an annual rate of \( 8\% \).
This means that each year, the investment accumulates more value, leading to exponential growth. This growth is not linear, meaning it doesn't just climb by a fixed amount every year. Instead, it builds upon the original investment plus the gains achieved in previous years. In other words, it's the interest on top of other interest, known as compound interest, resulting in a snowball effect.
Key Points:
Different factors can influence this growth, but one of the most common is the rate of return or the interest rate. In our exercise, we have an investment of \( \$1000 \) growing at an annual rate of \( 8\% \).
This means that each year, the investment accumulates more value, leading to exponential growth. This growth is not linear, meaning it doesn't just climb by a fixed amount every year. Instead, it builds upon the original investment plus the gains achieved in previous years. In other words, it's the interest on top of other interest, known as compound interest, resulting in a snowball effect.
Key Points:
- Investment growth is described by the formula \( A = P (1 + r)^t \), where \( P \) is the principal amount.
- The growth rate \( r \) at \( 8\% \) must be converted into a decimal (so \( 0.08 \)).
- Investment growth is usually exponential rather than linear.
Exponential Functions
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of investment growth, this is seen in the formula \( A = P(1 + r)^t \), where \( (1 + r) \) is the base raised to the exponent \( t \).
Unlike linear functions which increase at a constant rate, exponential functions increase more and more rapidly. The reason they grow at this accelerating pace is because they incorporate the effects of compounding. For our example, the base \( 1.08 \) represents the dollar value of the investment growing by \( 8\% \) annually.
Understanding exponential functions can help in predicting how investments develop over long periods:
Unlike linear functions which increase at a constant rate, exponential functions increase more and more rapidly. The reason they grow at this accelerating pace is because they incorporate the effects of compounding. For our example, the base \( 1.08 \) represents the dollar value of the investment growing by \( 8\% \) annually.
Understanding exponential functions can help in predicting how investments develop over long periods:
- The principal \( P = 1000 \) stays the same, but \( A \) (the future amount) grows dramatically over time because of the compounded elements in \( (1.08)^t \).
- This formula allows investors to calculate the future value of an investment at any point in time \( t \).
- It illustrates the power of compounding, with the growth accelerating each year.
Doubling Time
Doubling time is the period it takes for an investment to grow twice its original size. It's a practical measure to understand how fast your money will grow, which is crucial for planning and decision-making.
In finance, finding the doubling time involves solving for \( t \) in the equation where \( 2 = (1 + r)^t \). For our exercise, the calculation was done using logarithms because you're often dealing with exponential functions.
To solve for the amount of time it takes for the investment to double:
In finance, finding the doubling time involves solving for \( t \) in the equation where \( 2 = (1 + r)^t \). For our exercise, the calculation was done using logarithms because you're often dealing with exponential functions.
To solve for the amount of time it takes for the investment to double:
- Set \( A = 2000 \) in the equation, which represents double the initial \( 1000 \).
- Use the formula \( 2 = (1.08)^t \) and solve for \( t \) using logarithmic operations.
- The resulting calculation \( t \approx 9.006 \) shows it takes a little over 9 years to double \( \$1000 \) at an \( 8\% \) annual interest rate.
