Question

Simplify the expression if possible. $$\frac{4 x}{20}$$

Short answer

The simplified expression of \(\frac{4x}{20}\) is \(\frac{x}{5}\)

Step-by-step solution

Identify the GCD of the Numerator and Denominator

The Greatest Common Divisor (GCD) of 4 and 20 is 4.

Divide Numerator and Denominator by their GCD

Divide both the numerator and the denominator by their GCD, resulting in a new expression \(\frac{4/4 \cdot x}{20/4}\) = \(\frac{x}{5}\)

Final Simplified Expression

We end up with the simplified expression \(\frac{x}{5}\)

Key concepts

Greatest Common Divisor

When simplifying algebraic expressions, particularly fractions, identifying the Greatest Common Divisor (GCD) is a crucial first step. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

For instance, consider the numbers 4 and 20. Here's how you can find their GCD: list out the factors of each number. The factors of 4 are 1, 2, and 4. The factors of 20 are 1, 2, 4, 5, 10, and 20. The common factors are 1, 2, and 4, and the greatest of these is 4.

To utilize the GCD for simplifying fractions, divide the numerator and the denominator by the GCD. This process will reduce the fraction to its simplest form. It's a systematic way to ensure that the fraction is as simple as it can be without guessing or missing potential simplifications.

Understanding and calculating the GCD is an invaluable tool in algebra, and it's widely used in operations such as reducing fractions, simplifying algebraic fractions, and finding least common multiples for algebraic expressions.

Simplify Fractions

Simplifying fractions is an essential skill in algebra that allows for expressions to be presented in their most basic form. It requires dividing both the numerator (top number) and the denominator (bottom number) by their GCD.

Let's take the fraction \(\frac{4x}{20}\) from our exercise. The GCD of 4 and 20, as we discovered, is 4.

Divide to Simplify

By dividing both the numerator (4x) and the denominator (20) by 4, you obtain the simpler fraction \(\frac{x}{5}\). This process illustrates the straightforward path to simplification: divide both parts of the fraction by the same non-zero number.

Remember, when simplifying, the goal is not just to make the fraction smaller, it is to ensure that the numerator and denominator are as small as possible while still maintaining the same value of the fraction. Simplifying can make further algebraic operations much easier and can be a critical tool when solving equations, comparing fractions, or finding least common denominators.

Algebraic Fractions

Algebraic fractions are just like numerical fractions, but they include variables (like \(x\), \(y\), etc.) in their numerators, denominators, or both. Simplifying algebraic fractions uses the same principles we apply to numerical fractions. Identify the GCD, but this time including variables.

In our exercise with the algebraic fraction \(\frac{4x}{20}\), the variable \(x\) is involved. Even when variables are present, you can still find the GCD for the numerical coefficients and divide them accordingly. Here, the coefficient 4 of the variable \(x\) and the number 20 share a GCD of 4, as covered in previous examples.

Variables in Fractions

When dealing with variables, ensure the GCD only includes the numerical part. Leave the variable intact unless it's a common factor present in both the numerator and the denominator. In this case, the \(x\) in the numerator has no matching factor in the denominator, so it remains after simplification.

Understanding algebraic fractions is fundamental when dealing with algebraic equations, interpreting graphed functions, and solving real-world problems involving ratios and rates. Simplification can lead to clearer understanding, easier computations, and sometimes, the ability to recognize patterns within the algebraic expressions.