Problem 88 - College Algebra

Question

If \(\ S 4000\) is deposited into an account paying \(3 \%\) interest compounded annually and at the same time \(\ S 2000\) is deposited into an account paying \(5 \%\) interest compounded annually, after how long will the two accounts have the same balance?

Step-by-Step Solution

Answer

To find the time which makes the balance of both accounts equal, we need to evaluate the expression \[t = \frac{ln(2)}{ln(1 + 0.03) - ln(1 + 0.05)}\] This value, once evaluated, will indicate the number of years, maybe rounding to the nearest year for a complete figure.

Step 1: Understand the compound interest formula

First, we need to understand the formula for annually compounded interest which is \[A = P(1 + r/n)^{nt}\] where \(A\) is the amount of money accumulated after n years, including interest; \(P\) is the principal amount (the initial amount of money); \(r\) is the annual interest rate (in decimal); \(n\) is the number of times that interest is compounded per year; and \(t\) is the time in years.

Step 2: Write the expression for each account

Since both accounts are compounded annually, n is 1. For the first account, we have \[A_1 = 4000(1 + 0.03)^t\]. For the second account, we have \[A_2 = 2000(1 + 0.05)^t\]. We need to find the amount of time \(t\) when both accounts will have the same balance ('equal amounts'), so we need to set these two formulas equal to each other.

Step 3: Solve the equation

Setting the two expressions equal to each other, we solve the equation \[4000(1 + 0.03)^t = 2000(1 + 0.05)^t\] Simplify the equation by dividing both sides by 2000 to obtain \[(1 + 0.03)^t = 2(1 + 0.05)^t\] This is a specific form of exponential equations, where you can take the natural logarithm (ln) on both sides to solve for t. Taking log on both sides, we obtain \[t \cdot ln(1 + 0.03) = ln(2) + t \cdot ln(1 + 0.05)\] Rearranging the terms to one side gives \[t \cdot ln(1 + 0.03) - t \cdot ln(1 + 0.05) = ln(2)\] Factor out the t and solve for t \[t = \frac{ln(2)}{ln(1 + 0.03) - ln(1 + 0.05)}\]

Step 4: Evaluate the expression

The above expression will give the value of time in years when the balance in both the accounts will be the same. This value can be found either by using any mathematical software or by using the logarithm function on a calculator. After the actual calculation, the time obtained should be rounded to the nearest year.

Key Concepts

Exponential Equations

Exponential equations are types of equations where variables appear as exponents. They often emerge in financial calculations, such as compound interest problems. An example of an exponential equation is the compound interest formula:

• \( A = P(1 + r/n)^{nt} \)
• Here, \(A\) represents the total amount of money after interest, \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time in years.

Exponential equations frequently involve expressions of the form \((1 + r)^t\), indicating how the base amount \(P\) grows over time. To solve exponential equations like these, you often need to isolate the variable in the exponent. This can involve taking logarithms to simplify the equation, allowing you to resolve variables such as \(t\), the time in years required for specific financial goals to be met.

Annual Interest Rate

The annual interest rate is the percentage at which your investment or deposit grows each year. When an interest rate is described as compounded annually, it means that the interest is applied once every year. This frequently results in exponential growth, making the investment or deposit increase at a faster rate over time.

• For instance, in our original problem, the first account has a 3% annual interest rate, written as 0.03 in decimal form, while the second account has a 5% annual interest rate, written as 0.05 in decimal form.

Understanding the annual interest rate's role within the compound interest formula is crucial, as it affects how quickly your money grows. The rate is added to 1 and raised to the power of \(t\), indicating growth over each time period. The exponential nature of the equation highlights why even small differences in rates can lead to significant variations in outcomes over time.

Natural Logarithm

The natural logarithm (often abbreviated as ln) is a mathematical function that helps in solving exponential equations. It is crucial for dealing with equations where the unknown variable is an exponent. The natural logarithm is based on the constant \(e\), approximately equal to 2.718, which is a fundamental base for natural growth and decay processes.

• In exponential equations like \(4000(1 + 0.03)^t = 2000(1 + 0.05)^t\), taking the natural logarithm on both sides is a systematic method for simplifying the equation and isolating the variable \(t\).
• By applying natural logarithms, the exponents in the equation become coefficients, leading to a simpler algebraic form, such as \(t \cdot ln(1 + 0.03) = ln(2) + t \cdot ln(1 + 0.05)\).

As natural logarithms are integral in solving for variables within exponential growth or decay situations, mastering this concept is essential for any student keen on understanding compound interest and similar financial computations.