Problem 8 - Algebra for College Students

Question

Solve \(i=\operatorname{Prt}\) for \(P\), given that \(r=8 \frac{1}{2} \%, t=2\) years, and \(i=\$ 204\).

Step-by-Step Solution

Answer

The principal \( P \) is \( 1200 \).

Step 1: Understand the Formula

The formula given in the exercise is \( i = P imes r imes t \). Here, \( i \) is the interest earned, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time.

Step 2: Substitute Known Values

We need to solve for \( P \). Substitute the given values into the formula: \( i = 204 \), \( r = 8.5\% = 0.085 \), and \( t = 2 \). So, the equation becomes \( 204 = P \times 0.085 \times 2 \).

Step 3: Simplify the Equation

First, simplify the right side of the equation: calculate \( 0.085 \times 2 \). This results in \( 0.17 \). The equation simplifies to: \( 204 = P \times 0.17 \).

Step 4: Solve for P

To find \( P \), divide both sides of the equation by \( 0.17 \). This results in \( P = \frac{204}{0.17} \).

Step 5: Calculate the Division

Perform the division \( P = \frac{204}{0.17} \). This gives \( P = 1200 \).

Key Concepts

Interest Formula

When we talk about the simple interest formula, it's a handy way to understand how interest is calculated over time. The formula is given by \[ i = P \times r \times t \]where:

• \( i \) is the total interest earned or paid.
• \( P \) represents the principal, or the original amount of money.
• \( r \) is the annual interest rate (as a decimal).
• \( t \) stands for the time period the money is borrowed or invested, often expressed in years.

This formula is simple because it assumes that the interest is not compounded, meaning it does not get added to the principal over time. Instead, the interest is purely a product of multiplying the principal by the rate and time.

Solving Equations

Solving equations, especially in the context of simple interest, involves basic algebraic manipulation. Let's walk through it step by step. When you have an equation like \[204 = P \times 0.085 \times 2\]you start by simplifying what's on one side of the equation. This simplification gives us \[204 = P \times 0.17\],making it clearer and easier to handle.
To solve for \( P \), divide each side of the equation by \( 0.17 \). Mathematically, it looks like this:\[P = \frac{204}{0.17}\].
This division isolates \( P \), allowing you to calculate its value. In this scenario, \( P \) equals 1200, meaning the original principal amount that was earning or paying interest at the given rate and time.

Financial Mathematics

Financial mathematics encompasses various concepts like time value of money, interest rates, and investment returns. A core element is understanding how interest impacts money over time. Simple interest is foundational, reflecting how money grows linearly. When you use the \[ i = P \times r \times t \]formula, it shows how increments are added consistently, unlike compound interest, which increases by reinvesting the earnings.
Key skills in financial mathematics include:

• Understanding and applying interest formulas appropriately.
• Performing operations like solving, substituting, and simplifying equations.
• Recognizing the economic impact of various interest calculations over different periods.

These skills are crucial whether you are saving, investing, or managing loans, as they form the basis for understanding more complex financial concepts.