Problem 52 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

It will take 20 years for the principal to triple at 10% simple interest.

Step 1: Understanding Simple Interest

Simple interest means that interest is calculated only on the initial principal and not on any accumulated interest. The formula for calculating simple interest over time is given by:\[ I = P \cdot r \cdot t \]where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years.

Step 2: Set Up the Equation to Find Time

We need to find out how long it will take for the principal \( P \) to triple. If \( P \) triples, the total amount becomes \( 3P \). The interest earned after this time must be \( 2P \), because the total amount (\( 3P \)) is the principal \( P \) plus the interest \( 2P \). So, we set up the equation for simple interest as:\[ 2P = P \cdot 0.10 \cdot t \]

Step 3: Solve for Time \( t \)

Now, solve the equation \( 2P = P \cdot 0.10 \cdot t \) for \( t \). Divide both sides of the equation by \( P \) to simplify:\[ 2 = 0.10t \]Next, divide both sides by 0.10 to isolate \( t \):\[ t = \frac{2}{0.10} \]

Step 4: Calculate the Value of \( t \)

After simplifying \( t = \frac{2}{0.10} \), we calculate:\[ t = 20 \]So, it will take 20 years for the principal \( P \) to triple at \( 10\% \) simple interest.

Key Concepts

Financial Mathematics

Financial mathematics is all about using mathematical formulas to solve problems related to finance. You often use it when dealing with investments, loans, and savings accounts.
It’s a crucial tool for understanding how money can grow or shrink over time. In the case of simple interest, financial mathematics allows us to determine how much interest will be earned on a principal sum over a specific period.

• Understanding financial mathematics helps make informed decisions about available options.
• It is essential in budgeting, saving, and investing.

Simple interest is one of its simplest applications, focusing solely on the initial invested amount without considering any additional interest earned.

Interest Rate Calculation

Interest rate calculation is key to determining how much interest you will earn or owe over time. With simple interest, the rate is applied only to the original principal amount, making calculations straightforward.
For simple interest, the formula used is:\[ I = P \cdot r \cdot t \]where:

• \( I \) is the interest earned,
• \( P \) is the principal amount,
• \( r \) is the interest rate (as a decimal),
• \( t \) is the time in years.

This formula helps determine how much additional money will be gained from an investment or charged on a loan over time. Adjusting any of these variables allows you to calculate the others.

Solving Linear Equations

Solving linear equations is a common practice in mathematics, including problems involving simple interest. It involves finding the unknown variable, just like we did with the time variable \( t \) in the simple interest equation.
Here’s how to tackle a basic linear equation step-by-step:

• First, set up your equation based on the problem (e.g., \( 2P = P \cdot 0.10 \cdot t \)).
• Simplify the equation by performing operations that make it easier to solve (e.g., dividing both sides by \( P \)).
• Isolate the variable of interest by using algebraic operations (e.g., dividing both sides by the coefficient of \( t \), 0.10).
• Calculate the result to find the value of the unknown (e.g., \( t = 20 \)).

By breaking down the equation into manageable steps, you make it easier to solve and understand.