Problem 50 - Prealgebra

Question

Solve each equation. $$\frac{x}{45}=\frac{8}{18}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 20 \).

Step 1: Cross-multiply

To eliminate the fractions, we will cross-multiply. The given equation is \( \frac{x}{45} = \frac{8}{18} \). By cross-multiplying, we have: \( 18x = 8 \times 45 \).

Step 2: Multiply and Simplify

Now, we need to multiply 8 by 45. \( 8 \times 45 = 360 \). So, the equation 18x = 360 comes from this step.

Step 3: Solve for x

To find \( x \), divide both sides of the equation by 18. \( x = \frac{360}{18} \).

Step 4: Simplify the Solution

Simplify \( \frac{360}{18} \) to find \( x \). Dividing 360 by 18, we get \( x = 20 \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a useful technique for solving equations that involve fractions. It allows you to eliminate the fractions by multiplying both sides of the equation. This is particularly handy when you have a fraction on both sides of the equation. In our example, we have \( \frac{x}{45} = \frac{8}{18} \). To cross-multiply, you multiply the numerator (the top number) of the first fraction by the denominator (the bottom number) of the second fraction and vice versa:

• First, multiply 18 (denominator of the second fraction) by \( x \) (numerator of the first fraction): \( 18 \times x \).
• Then, multiply 45 (denominator of the first fraction) by 8 (numerator of the second fraction): \( 45 \times 8 \).

This gives us the equation \( 18x = 8 \times 45 \). This process effectively removes the fractions from the equation, making it easier to solve.

Fractions

Fractions are numbers that express one quantity as a part of another quantity. They consist of a numerator (the top number) and a denominator (the bottom number). In equations, fractions are used to show how many parts of a whole we have. When dealing with equations such as \( \frac{x}{45} = \frac{8}{18} \), fractions can be intimidating. However, understanding the relationship between the numerators and denominators can simplify the equation-solving process. Ensuring the fractions are compared correctly is key. Familiarity with operations like cross-multiplication can turn fractions into more manageable expressions.If learners understand the components and basic operations involving fractions, they will find such equations less challenging.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).For instance, after cross-multiplying in our example, we had the equation \( 18x = 360 \). By dividing both sides by 18, we arrive at the fraction \( \frac{360}{18} \), which needs simplifying. The GCD of 360 and 18 is 18. When both the numerator and the denominator are divided by 18, the simplified form is 20. Here's how you can simplify step-by-step:

• Identify the GCD of 360 and 18, which is 18.
• Divide both numerator and denominator by 18: \( \frac{360}{18} = 20 \).

Simplifying fractions helps express them in the most straightforward form, making mathematical solutions clearer and easier to interpret.

Prealgebra

Prealgebra is an introductory stage in mathematics where students become familiar with the fundamental concepts that pave the way for algebra. It focuses on arithmetic operations, basic properties of numbers, and foundational concepts like solving equations with fractions. Understanding prealgebra is crucial as it helps build confidence and prepares students for more complex mathematical problem-solving. It emphasizes concepts such as cross-multiplication, fractions, and arithmetic operations needed to simplify expressions. In the given example, prealgebra involves:

• Cross-multiplying to remove fractions from an equation.
• Using arithmetic operations to solve for the variable \( x \).
• Simplifying solutions to achieve final results.

By mastering these prealgebraic skills, students are better equipped to tackle algebra and other higher-level mathematics with a solid foundation.