Problem 34 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{60 x^{2}}{36 x}$$

Step-by-Step Solution

Answer

The fraction \(\frac{60x^2}{36x}\) reduces to \(\frac{5x}{3}\).

Step 1: Identify Common Factors

First, we need to find the greatest common factor (GCF) of the numerator and the denominator. The numerator is \(60x^2\) and the denominator is \(36x\). To do this, we look at both the numerical coefficients and the variables. The GCF of the numbers 60 and 36 is 12. For the variable part, the common factor is \(x\). So the GCF of the expression is \(12x\).

Step 2: Divide by the GCF

We will divide both the numerator and the denominator by the GCF \(12x\). For the numerator, \(\frac{60x^2}{12x} = 5x\). For the denominator, \(\frac{36x}{12x} = 3\). So, after dividing, the fraction becomes \(\frac{5x}{3}\).

Step 3: Simplify the Fraction

Now, we check if the fraction \(\frac{5x}{3}\) is in its simplest form. There are no common factors between 5 and 3, and the variable \(x\) already exists only in the numerator, so \(\frac{5x}{3}\) is indeed simplified.

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a key concept in mathematics that helps simplify fractions by identifying the largest factor shared by two numbers or expressions. When we look at a numeric fraction, the GCF is the biggest number that can divide both the numerator and the denominator without leaving any remainder.
In an algebraic context, like our exercise with variables, the GCF comes into play with both coefficients and the literal variables. For the expression \(60x^2\) and \(36x\), we first consider the numbers: 60 and 36. The GCF of 60 and 36 is 12 because it’s the largest number that divides both evenly.
Variables are treated similarly. In this case, the variable part of the expression has \(x^2\) in the numerator and \(x\) in the denominator, making \(x\) the common variable factor.
So, the GCF of \(60x^2\) and \(36x\) is \(12x\). Finding this factor is crucial because it allows us to reduce the fraction by cancelling out the largest common elements.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about making expressions easier to work with by reducing them to their simplest form. This involves the process of finding common factors and reducing the fractions, which is directly linked to our previous discussion about the GCF. Once you find the GCF, you divide both the numerator and the denominator by that factor.
Let's consider our example. By identifying \(12x\) as the GCF of \(60x^2\) and \(36x\), we divide both parts of the fraction by this factor:

• The numerator becomes \(\frac{60x^2}{12x} = 5x\).
• The denominator becomes \(\frac{36x}{12x} = 3\).

These steps simplify the fraction from \(\frac{60x^2}{36x}\) to \(\frac{5x}{3}\).
Checking for further simplification involves verifying whether any other common factors exist in the new fraction's terms, which they do not in this case.

Variables in Fractions

Variables in fractions can sometimes present challenges, but they follow the same rules as numbers regarding operations like simplifications and cancellations.
When variables appear in both the numerator and the denominator, like in our exercise \(\frac{60x^2}{36x}\), they can be simplified by cancelling out similar terms.
It’s crucial to observe how variables behave during simplification:

• The variable \(x^2\) in the numerator implies two occurrences of \(x\).
• The denominator has one \(x\) factor.

When simplifying, cancel out one \(x\) from both the numerator and the denominator, leaving \(5x\) in the numerator and 3 in the denominator.
This systematic cancellation helps break down what seems complex into understandable parts. So, after cancelling the common \(x\) and dividing the coefficients by their GCF, the expression \(\frac{60x^2}{36x}\) neatly simplifies to \(\frac{5x}{3}\). This fraction is now in its simplest form because no further factors are shared between the numerator and the denominator.