Problem 40 - Elementary Algebra Within Reach

Question

Solve the proportion. $$\frac{2}{4.5}=\frac{t}{0.5}$$

Step-by-Step Solution

Answer

The solution to the proportion is \(t = \frac{1}{4.5}\).

Step 1: Setting up the equation

We have the proportion \(\frac{2}{4.5}=\frac{t}{0.5}\). To solve for \(t\), we will use cross multiplication, which involves multiplying the extremes (2 and 0.5) and the means (4.5 and \(t\)).

Step 2: Cross multiply

Upon cross-multiplying the equation becomes \(2 \times 0.5 = 4.5 \times t\). This simplifies to \(1 = 4.5t\).

Step 3: Solve for t

To isolate \(t\), divide both sides of the equation by 4.5. The result is \(t = \frac{1}{4.5}\).

Key Concepts

Cross Multiplication

Cross multiplication is a powerful tool used to solve proportions, which are equations that express the equality of two ratios. Imagine you have a fraction on either side of an equals sign. In cross multiplication, you multiply the numerator on one side by the denominator on the other side, and vice versa. This method is particularly useful because it eliminates the fractions and allows for simpler equations.
For example, consider the proportion given in the exercise

• Original proportion: \(\frac{2}{4.5} = \frac{t}{0.5} \)
• Cross multiplied: \(2 \times 0.5 = 4.5 \times t \)
• Simplified equation: \(1 = 4.5t \)

As shown, cross multiplication transforms the problem from dealing with complex fractions to a straightforward linear equation, making it much easier to solve.

Linear Equations

Linear equations are mathematical statements that describe a straight line when graphed. They generally take the form \(ax + b = c\), where \(x\) is the variable, and \(a\), \(b\), and \(c\) are constants. Solving linear equations requires finding the value of the variable that makes the equation true.
In the exercise:

• After cross-multiplying, we transform the original proportion into the linear equation: \(1 = 4.5t\)

Our goal is to isolate the variable "\(t\)". Here, \(t\) is multiplied by 4.5, which means we should do the opposite operation (division) to solve for \(t\). So, we divide both sides by 4.5, giving us \(t = \frac{1}{4.5}\). Linear equations are foundational in algebra because they provide a straightforward process to find unknown variables.

Mathematical Fractions

Mathematical fractions represent a part of a whole and consist of two parts: the numerator (top number) and the denominator (bottom number). They are essential in expressing and working with portions of whole numbers or quantities. Fractions are involved in a wide array of mathematical concepts and operations, from basic arithmetic to solving proportions.
In the provided exercise:

• The fraction \(\frac{2}{4.5}\) is equal to \(\frac{t}{0.5}\).
• This proportion suggests that each side of the equation represents the same value when reduced to a unit fraction.

Working within the framework of fractions often requires converting them to common terms or manipulating them for simplification. In our scenario, using the concept of cross multiplication allows us to resolve the proportion by treating the fractions more manageably, leading to a successful solution of the problem.