Problem 13 - Elementary Algebra

Question

Solve each of the following problems. Keep in mind the suggestions we offered in this section. How long will it take $$\$ 4000$$ to double itself if it is invested at \(8 \%\) simple interest?

Step-by-Step Solution

Answer

It takes 12.5 years for the investment to double.

Step 1: Understand Simple Interest Formula

The formula for simple interest is \( I = P \cdot r \cdot t \), where \( I \) is the interest, \( P \) is the principal amount, \( r \) is the rate of interest per year, and \( t \) is the time in years. We want the interest \( I \) to be equal to the principal \( P \) since the amount doubles.

Step 2: Set Up the Doubling Condition

For the money to double, \( I = P = 4000 \). So, the total amount after interest is \( 2P = 2 \times 4000 \). Thus, we want \( 4000 = 4000 \cdot 0.08 \cdot t \).

Step 3: Solve for Time \( t \)

Substitute the known values into the formula: \( 4000 = 4000 \cdot 0.08 \cdot t \). This simplifies to \( 1 = 0.08t \). Solve for \( t \) by dividing both sides by 0.08: \( t = \frac{1}{0.08} \).

Step 4: Calculate the Time

Calculate \( t = \frac{1}{0.08} = 12.5 \). Thus, it will take 12.5 years for the investment to double.

Key Concepts

Interest Calculation

When you hear about simple interest, it’s all about calculating how much extra money an investment will earn over time. This is different from compound interest, but we'll focus on the simple one here. The straightforward formula helps you determine the interest by multiplying three components together: the principal (P), the interest rate (r), and the time (t). This formula is written as follows:

• \( I = P \cdot r \cdot t \)

In our given problem, the principal is \( \$4000 \), the interest rate is \(8\%\) or \(0.08\) in decimal form, and we are trying to find out the time it takes for the investment to double.
Transforming the interest into a usable form, e.g., decimals, makes calculations easier. This calculation only considers the initial amount without adding any future interest from previous periods. That's why it's called simple interest.

Doubling Time

Doubling time is a popular concept in finance, representing the duration required for an investment to become twice its original value. In simple interest scenarios, we want to know when the money gained from interest (I) equals the initial amount or principal (P).
For the given problem:

• Principal (P) = \( \\(4000 \)
• Interest earned (I) = \( \\)4000 \)
• Total amount after interest = \(2P\)

This setup leads us to understand that when the interest equals the principal, you've doubled your money. We already set up our equation \( 4000 = 4000 \cdot 0.08 \cdot t \) to explore how long this process takes. Doubling time offers a clear perspective to investors about the timeline required to reach specific financial targets.

Investment Growth

Investment growth through simple interest is linear, not exponential. Thus, your money will grow at a steady rate over time. This rate is decided by the interest percentage, which in our case is \(8\%\).
Every year, your investment will increase by a fixed amount calculated using the simple interest formula. However, unlike compound interest, which reinvests earned interest, simple interest only processes the principal for interest calculations.
Using the initial investment, you can predict future values like this:

• Each year, earn \(\\(4000 \times 0.08 = \\)320\) as interest.
• The amount after a year becomes \(\\(4000 + \\)320\).
• This continues until the growth doubles the initial investment.

Knowing this helps in planning financial goals, as you can see how each year contributes to the final amount.

Algebraic Solution

Algebra comes into play when solving financial questions, like calculating the doubling time of an investment with simple interest. Here's how it works:
First, establish the equation from the scenario where interest equals the principal:

• \( 4000 = 4000 \cdot 0.08 \cdot t \)

The next step involves simplification and solving for \(t\):

• Divide both sides by \(4000\) to isolate \(t\): \(1 = 0.08t\).
• Divide by the rate \(0.08\) to clear the terms from \(t\): \(t = \frac{1}{0.08}\).
• Calculate the value: \(t = 12.5\) years.

Through these algebraic steps, we uncover the solution, showing that it would take 12.5 years for our \(\$4000\)investment to double at an \(8\%\) simple interest rate.