Problem 1 - Elementary Algebra

Question

For Problems \(1-12\), solve each of the equations. These equations are the types you will be using in Problems 13-40. $$ 950(0.12) t=950 $$

Step-by-Step Solution

Answer

The solution is \(t = \frac{475}{57}\).

Step 1: Simplify the Equation

The given equation is \(950(0.12)t = 950\). First, simplify the expression on the left-hand side by multiplying 950 by 0.12. This gives us \(950 \times 0.12 = 114\), resulting in the equation \(114t = 950\).

Step 2: Isolate the Variable

To isolate \(t\), divide both sides of the equation by 114. This gives us \(t = \frac{950}{114}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{950}{114}\) by finding the greatest common divisor (GCD) of 950 and 114 and dividing both the numerator and denominator by this GCD. The GCD is 2, so the fraction simplifies to \(\frac{475}{57}\). This is the simplest form of the fraction.

Key Concepts

Fractions

Fractions are a way to express parts of a whole using two numbers: a numerator and a denominator. In the context of solving algebraic equations, fractions often show up when you isolate a variable by dividing both sides of an equation. This occurs because not all division results in a whole number.

For example, in the equation resulting from the simplification step, we had found that \(t = \frac{950}{114}\). Here, 950 is the numerator, and 114 is the denominator. The fraction represents the exact value of \(t\) when solving the equation.

• Numerator: the top part of the fraction, indicating how many parts we have.
• Denominator: the bottom part, indicating the size of each part or the total number of equal parts the whole is divided into.

In many cases, we need to simplify fractions to their lowest terms to make the numbers easier to interpret and work with, such as in the final simplification result \(\frac{475}{57}\).

Understanding fractions is crucial since they frequently appear in algebra, especially when equations do not yield integers.

Simplification

Simplification is the process of making an equation or expression easier to work with or understand. It involves reducing expressions to their simplest form. This can make calculations more manageable and help in identifying the structure or pattern of the mathematical situation.

In the given problem, simplification happens in a few ways:

• Arithmetic Simplification: Initially, we simplify the left-hand side of the equation by performing the multiplication. So, \(950 \times 0.12\) becomes 114, leading to the simplified equation \(114t = 950\).
• Fractions Simplification: After isolating the variable, \(t\) was expressed as a fraction \(\frac{950}{114}\). Simplifying this requires finding the greatest common divisor (GCD), which was 2 in this case. By dividing both the numerator and denominator by the GCD, the fraction became \(\frac{475}{57}\).

Remember, simplifying early in the problem-solving process can help prevent errors later and makes subsequent steps more straightforward. It also helps to present the equation in a more understandable and concise form.

Variables

Variables are symbols used to represent numbers in mathematical expressions and equations. They allow us to write general formulas and equations that hold true for many situations. In algebra, we often solve for these variables to find unknown quantities.

For example, the variable \(t\) in the given equation \(114t = 950\) is what we're solving for. The goal is to determine the value of \(t\) that makes the equation true.

When dealing with variables, it's important to:

• Isolate the Variable: The primary aim is to get the variable on one side of the equation. In our problem, this was achieved by dividing both sides by 114.
• Literal Coefficients: Understand how coefficients (the numbers multiplying the variable) affect the variable. Here, 114 is the coefficient of \(t\).

Variables provide flexibility and power in algebra, letting us express relationships between quantities and solve complex problems by focusing on what is unknown. Always remember that isolating the variable is key to solving equations effectively.