Problem 50 - Algebra for College Students

Question

Solve each of Problems \(47-62\) by setting up. How long will it take \(\$ 700\) to triple itself at \(10 \%\) simple interest?

Step-by-Step Solution

Answer

It will take 20 years for the money to triple.

Step 1: Understand Simple Interest Formula

The formula for simple interest is given by the equation: \[ I = P \times r \times t \] where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the rate of interest per year, and \( t \) is the time in years. Our goal is to find \( t \), the time needed for the principal to triple.

Step 2: Determine Final Amount

Since the problem states that the initial amount \( P \) triples, the final amount \( A \) is \( 3 \times P \). Given \( P = 700 \), the final amount is \( 3 \times 700 = 2100 \).

Step 3: Calculate Interest Earned

The interest earned \( I \) can be found by subtracting the principal from the final amount: \[ I = A - P = 2100 - 700 = 1400 \]

Step 4: Rearrange the Simple Interest Formula

To find \( t \), we rearrange the simple interest formula for \( t \): \[ t = \frac{I}{P \times r} \]

Step 5: Substitute Known Values

Substitute the known values into the formula: \( I = 1400 \), \( P = 700 \), and \( r = 0.10 \) (since 10% as a decimal is 0.10): \[ t = \frac{1400}{700 \times 0.10} \]

Step 6: Solve for Time \( t \)

Calculate \( t \) as follows: \[ t = \frac{1400}{70} = 20 \] Thus, it will take 20 years for the \( \$ 700 \) to triple at 10% simple interest.

Key Concepts

Interest Calculation

Interest calculation is fundamental when dealing with financial topics. The formula for simple interest is \[ I = P \times r \times t \]where:

• \( I \) is the interest earned,
• \( P \) is the principal or initial amount,
• \( r \) represents the annual interest rate, and
• \( t \) stands for the time period, typically measured in years.

In simple interest, the amount is applied to the principal amount only, making the calculation straightforward.
This simplicity is why it's often used for educational purposes or short-term loans where the amount of interest does not compound.

Principal Amount

The principal amount is the initial sum of money that is invested or borrowed. It is a critical component of the simple interest formula.
In our original exercise, we had a principal amount of \\(700. The principal remains fixed throughout the duration
of the period considered, meaning the interest earned over time depends on this original amount.
The goal of our exercise was to determine how long it would take for the principal, \\)700, to triple itself<
using simple interest, not changing this principal amount over time.
This fixed nature of the principal makes simple interest an attractive choice for straightforward and short-term calculations.

Interest Rate

The interest rate is usually expressed as a percentage.
In simple interest calculations, this rate is applied annually to the principal amount.
For instance, a 10% interest rate means that the investor earns 10% of the principal as interest each year.

• Interest rates significantly influence the amount of interest earned, making it a critical factor in financial decisions.
  
• Even a small change in interest rates can result in substantial differences in the amount of interest accrued, especially for larger principal amounts or longer time periods.
  

For the example exercise, a rate of 10% was used.
This means that every year, 10% of the principal amount is added as interest over the duration considered.
Always convert percentage rates to their decimal form when using equations (e.g., 10% becomes 0.10).

Time Calculation

Time calculation is essential to find out how long it takes for the principal to reach a desired financial goal, such as tripling in the given exercise.
Rearranging the simple interest formula allows us to solve for time \( t \): \[ t = \frac{I}{P \times r} \]

• This formula provides the duration in years that it will take for an initial sum to reach a specific goal.
  
• In our example, knowing the interest earned \( I = 1400 \), principal \( P = 700 \), and the rate \( r = 0.10 \), we calculated \( t = 20 \) years.
  
• This means it will take 20 years for \$700 to triple at a 10% simple interest rate.

Careful calculation and understanding of how time impacts the growth of interest are crucial for any financial plan
or investment strategy using simple interest.