Problem 14 - Elementary Algebra

Question

How many years will it take $$\$ 1000$$ to double itself if it is invested at \(5 \%\) simple interest?

Step-by-Step Solution

Answer

It will take 20 years.

Step 1: Understand Simple Interest Formula

The simple interest formula is given by \( I = P \times r \times t \), where \( I \) is the interest, \( P \) is the principal (initial amount), \( r \) is the rate of interest per year, and \( t \) is the time in years.

Step 2: Set Up the Equation for Doubling

We want the investment to double, meaning the final amount should be \( 2 \times P = 2 \times 1000 = 2000 \). The interest earned when doubling the principal is therefore \( 1000 \), because \( 2000 - 1000 = 1000 \).

Step 3: Substitute Known Values into the Formula

Substitute the known values into the simple interest formula: \( 1000 = 1000 \times 0.05 \times t \).

Step 4: Solve for Time \( t \)

Simplify the equation to find \( t \). First, divide both sides by \( 1000 \) to get \( 1 = 0.05 \times t \). Then, divide by \( 0.05 \) to isolate \( t \): \( t = \frac{1}{0.05} \).

Step 5: Calculate \( t \)

Calculate \( t \) by dividing \( 1 \) by \( 0.05 \): \( t = 20 \).

Key Concepts

Doubling Time

Understanding the concept of doubling time is crucial when dealing with investments and interest rates. Doubling time refers to the period it takes for an investment or a sum of money to grow to twice its size. This involves calculating how long an initial amount will take to increase to double the value under a specific interest rate.

In our example, we started with \(\\(1000\) and want to find out how long it will take to become \(\\)2000\) at a \(5\%\) simple interest rate.

• The original principal is \(\\(1000\).
• The goal is to grow this amount to \(\\)2000\).
• The interest rate is \(5\%\) or \(0.05\) in decimal.

The focus here is on using the simple interest formula to determine how many years it takes for this to occur.

Simple Interest Formula

The simple interest formula is a straightforward way to calculate interest on an investment or loan. The formula is given by:\[ I = P \times r \times t \]Where:

• \(I\) is the interest earned or paid.
• \(P\) is the principal amount, or initial investment/loan.
• \(r\) is the rate of interest per year (expressed as a decimal).
• \(t\) is the time the money is invested or borrowed for, in years.

To illustrate, consider you invest \(\$1000\) at an interest rate of \(5\%\). Over time, this formula can help you calculate the amount of interest your principal will earn. This cumulative interest determines when your initial investment will double.

Interest Calculation

Interest calculation under simple interest involves substituting known values into the formula to find unknowns like time or total interest. In the case of our \(\\(1000\) investment at a \(5\%\) interest rate, the interest earned should equal the principal for it to double.

This is set up in the equation:\[ 1000 = 1000 \times 0.05 \times t \]Here:

• The left side \(1000\) is the interest we want to earn.
• The \(1000\) on the right side is the initial investment.
• \(0.05\) is the interest rate per year.
• \(t\) is the number of years we want to find.

Solving for \(t\), we divide both sides by \(1000\) to simplify:\[ 1 = 0.05 \times t \]Then divide by \(0.05\):
\[ t = \frac{1}{0.05} = 20 \]This tells us that at a 5% simple interest rate, it will take 20 years for \(\\)1000\) to double to \(\$2000\). This calculation demonstrates how understanding simple interest can aid in making informed financial decisions.