Problem 16 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{4 x}{20}$$

Step-by-Step Solution

Answer

The simplest form of the given expression \(\frac{4x}{20}\) is \(\frac{x}{5}\).

Step 1: Identify key terms

We are provided with a fraction \(\frac{4x}{20}\) where the numerator is 4x and the denominator is 20.

Step 2: Reducing common factors

In this case, 4 is a common factor in both the numerator and the denominator. Therefore, we can divide the numerator and denominator by this common factor, which will simplify the expression.

Step 3: Perform the division

Divide the numerator 4x by 4, we get x. Divide the denominator 20 by 4, we get 5.

Step 4: Write the Simplified Expression

The simplified form after reducing common factors is then \(\frac{x}{5}\).

Key Concepts

Common Factors

Fractions are simplified by identifying common factors in the numerator and the denominator. A common factor is a number that divides both the numerator and the denominator evenly.
The process of finding

• Look at each number in the fraction. In our example [\(\frac{4x}{20}\)], we check the numbers 4 and 20.
• Consider the number 4. It divides itself evenly since \(\frac{4}{4} = 1\), and it also divides 20 evenly, because \(\frac{20}{4} = 5\). That means 4 is a common factor.

By dividing both the numerator and the denominator by their common factor, we simplify the fraction while maintaining its value. Remember, all fractions should be simplified to make solving equations easier.

Numerator

In a fraction, the numerator is the number located above the fraction line. It represents how many parts of a whole you have.
For example, in \(\frac{4x}{20}\), the numerator is 4x. When simplifying, our goal is to reduce this number by dividing it by a common factor.

• Start by identifying the whole number component of the numerator. In this case, it’s 4.
• Check for variable factors if present. Here, we have 'x', which is not simplified in this step.
• Recognize repeated factors. The number 4 is repeated in both the numerator and denominator.
• Perform the division. When we divide 4x by the common factor 4, we are left with \(x\).

Understanding the role of the numerator helps to simplify expressions correctly and ensures accuracy in calculations.

Denominator

The denominator is the number below the fraction line. It indicates into how many pieces the whole is divided.
In \(\frac{4x}{20}\), the denominator is 20. Just like the numerator, we aim to simplify it using a common factor.

• Identify the denominator: It is the number 20 in our example.
• Find common factors with the numerator. Here, the common factor is 4.
• Simplify the denominator by dividing it by 4, giving us 5.

When the denominator is simplified, the entire fraction is equivalent in value but is easier to read and work with. Understanding the denominator's role is crucial for simplifying and comparing fractions efficiently.
This step is key in making math problems clearer and quicker to solve.