Problem 21 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{36 x}{27 x}$$

Step-by-Step Solution

Answer

The simplified form of the fraction \(\frac{36 x}{27 x}\) is \(\frac{4}{3}\).

Step 1: Identify common factors

To simplify this fraction, first identify any common factors in the numerator and denominator. Both 36 and 27 have a common factor of 9 and, the variable \(x\) is present both in the numerator and the denominator. So, 9 and \(x\) are the common factors.

Step 2: Cancel out common factors

Now cancel out the common factors. Divide 36 and 27 by 9, and also cancel out \(x\) from the numerator and denominator. This gives us \(\frac{36 \div 9 \ x}{27 \div 9 \ x}\) which simplifies to \(\frac{4}{3}\).

Step 3: Write the final simplified fraction

The simplified form of the fraction \(\frac{36 x}{27 x}\) is \(\frac{4}{3}\).

Key Concepts

Common Factors in Algebra

When dealing with algebraic expressions, identifying common factors is a vital skill. A common factor is a number or algebraic term that divides into two or more numbers or terms without leaving a remainder. In the context of algebra, these factors can be numerical as well as variable terms. For instance, in the expression \(\frac{36 x}{27 x}\), both the numerator (36x) and the denominator (27x) share numerical and variable factors. The number 9 is a common numerical factor since it divides evenly into both 36 and 27. Similarly, the variable \(x\) is a common factor because it appears in both terms. Recognizing these shared elements is the first step toward simplification.

Identifying common factors is particularly helpful when simplifying fractions. The goal is to break down the terms into their simplest form, which often makes algebraic operations clearer and more manageable. By reducing expressions with common factors, we streamline complex equations and make the relationships between terms more apparent. As you encounter algebraic expressions, look for numbers and variables that appear in all the parts of the expression you're working with. Once found, you're ready to move on to the process of cancelling these common factors.

Cancelling Common Factors

Cancelling common factors is akin to reducing a fraction in basic arithmetic. Once you've identified the common factors in an algebraic expression, you can 'cancel' them, which means to divide them out of the numerator and denominator. It's important to remember that you can only cancel factors that are multiplied together, not those that are added or subtracted.

For the expression \(\frac{36 x}{27 x}\), both 9 and \(x\) were identified as common factors in our previous section. To cancel them, you divide both the numerator and the denominator by the common numerical factor, 9, and also remove the variable \(x\) from both since it is the same on top and bottom. What remains is the simplified expression \(\frac{4}{3}\) without the \(x\), since anything divided by itself is 1 (except for zero).
Keep in mind that any term or variable that is left in the expression after cancelling the common factors cannot be reduced further unless there are additional common factors. Always check your final expression to ensure that no further simplification is possible.

Simplifying Fractions in Algebra

Simplifying fractions in algebra involves reducing them to their lowest terms by eliminating any common factors from the numerator and the denominator. It's a crucial step towards solving equations and understanding the relationships between variables.

To simplify an algebraic fraction like \(\frac{36 x}{27 x}\), apply the following steps after identifying and cancelling common factors:

• Divide both the numerator and the denominator by their greatest common factor (GCF).
• If there are variables in the fraction, cancel them out if they are the same in the numerator and the denominator.
• Ensure that the remaining expression cannot be reduced any further by checking for additional common factors.
• Write down the simplified fraction, which is the result you sought from the beginning.

In our example, after cancelling the common factor of 9 and the variable \(x\), we are left with the simplified fraction \(\frac{4}{3}\). This process does not just make the fraction easier to handle but also prepares you for solving more complex equations. Simplifying algebraic fractions paves the way for understanding how variables relate within an equation, laying the groundwork for more advanced topics such as solving for unknowns and working with polynomial expressions.