Problem 22 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{6 x}{10 x}$$

Step-by-Step Solution

Answer

The fraction \(\frac{6x}{10x}\) reduced to lowest terms is \(\frac{3}{5}\).

Step 1: Identify Common Factors

First, identify the common factors in the numerator and the denominator. The numerator is \(6x\) and the denominator is \(10x\). Both 6 and 10 are divisible by 2, and both the numerator and the denominator contain the variable \(x\).

Step 2: Divide by the Greatest Common Factor

The greatest common factor of the numbers 6 and 10 is 2. Divide both the coefficient and the variable \(x\) from the numerator and denominator. \[\frac{6x}{10x} = \frac{6 \div 2 \cdot x \div x}{10 \div 2 \cdot x \div x}\] Simplifying each part gives: \[\frac{3}{5}\] since \(x \div x = 1\).

Step 3: Verify the Result

Ensure that the fraction \(\frac{3}{5}\) is the simplest form. 3 and 5 have no common factors other than 1, which confirms the fraction is fully simplified.

Key Concepts

Understanding Lowest Terms

When a fraction is reduced to its lowest terms, it means that there are no common factors (other than 1) between the numerator and the denominator. This essentially simplifies the fraction while keeping its value unchanged. Let's break this down further:

• A fraction like \(\frac{6x}{10x}\) initially contains both numbers and variables. Our goal is to simplify it by reducing both parts of the fraction to their simplest forms.
• By finding a common factor and dividing both the numerator and denominator by this factor, what remains is a fraction in its lowest terms.
• This process ensures that the fraction is in its simplest state, making calculations easier and the expression cleaner.

Think of finding the lowest terms as cleaning up a mathematical expression. It reduces clutter and makes it readily understandable. In simpler fractions, this approach helps in streamlining algebraic manipulations in further calculations.

Finding the Greatest Common Factor (GCF)

The greatest common factor (GCF) is key in simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator. Here's how you can effectively determine and use it:

• First, list the factors of each number in the numerator and denominator. For example, the factors of 6 are 1, 2, 3, and 6; for 10, they are 1, 2, 5, and 10.
• The largest factor that appears in both lists is the GCF. In our case, it is 2.
• Once identified, divide both the numerator and denominator by this GCF to simplify the fraction.

Why is GCF important? Without using the GCF, fractions might not be reduced completely, which could complicate further calculations involving these fractions. Using the GCF ensures that the fraction is as simple as possible from the outset.

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra. It involves using the GCF to reduce a fraction to its simplest form. Let's dive into the step-by-step process:

• Identify the numerator and denominator of the fraction, for example, \(6x\) and \(10x\) respectively.
• Find the GCF of the numerical parts. As discussed, it is 2 in this scenario.
• Divide both the numerical coefficients and any common variables by this GCF. For the fraction \(\frac{6x}{10x}\), we divide both by 2, simplifying it to \(\frac{3}{5}\).
• Cancel out any variables that appear in both the numerator and the denominator, like \(x\) to obtain the simplest form.

Simplifying doesn't change the value of the fraction. It just makes working with it easier. This is particularly useful when dealing with algebraic expressions, as it makes solving equations or comparing fractions much more straightforward.