Problem 26 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (lesson \(4-4\) ) $$\frac{30 x^{2}}{51 x y}$$

Step-by-Step Solution

Answer

\( \frac{10x}{17y} \) is the simplest form.

Step 1: Identify and Factor Numerator and Denominator

The given fraction is \( \frac{30x^2}{51xy} \). Start by identifying and factoring both the numerator and the denominator. The numerator \(30x^2\) can be factored as \(2 \times 3 \times 5 \times x \times x\). The denominator \(51xy\) factors into \(3 \times 17 \times x \times y\).

Step 2: Cancel Common Factors

Check for common factors in both the numerator and the denominator. We see that both the numerator and the denominator have the factor \(3\) and \(x\). Cancel these common factors. This gives us \( \frac{10x}{17y} \).

Step 3: Check for Further Simplification

Look at the resulting fraction \( \frac{10x}{17y} \). The numerator and denominator contain no other common factors, and cannot be simplified further since 10, 17, and the variables do not share additional common factors.

Key Concepts

Factoring in Algebra

Factoring in algebra involves breaking down numbers or algebraic expressions into their simplest parts. It's a fundamental skill that helps simplify fractions or solve equations. This process is like finding out what numbers you can multiply together to get another number.

To factor an algebraic expression, identify every component that creates it. For instance, consider the expression \(30x^2\). You start by finding all prime numbers and factors involved:

• The number 30 can be broken down into prime factors: \(2 \times 3 \times 5\).
• For the variable part \(x^2\), it results in \(x \times x\).

Putting it together, you get \(2 \times 3 \times 5 \times x \times x\). This is factoring the numerator.

By doing the same for 51xy, you recognize that 51 equals \(3 \times 17\) and along with the variables you have \(3 \times 17 \times x \times y\). Understanding factoring gives you the tools to dissect and simplify complex expressions, making it a vital part of algebra.

Common Factors

Identifying common factors is a crucial step in simplifying fractions. A common factor is a number or algebraic component that divides exactly into both the numerator and the denominator of a fraction.

In mathematics, these are the shared components that can "cancel out" when simplifying. By removing common factors, you reduce the fraction to a simpler form. It's similar to realizing both parts of a fraction have been influenced by the same elements.

Consider the fraction \(\frac{30x^2}{51xy}\) from the example. After factoring, you can spot the common holders:

• Number 3 appears in both \(30x^2\) and \(51xy\).
• The variable x is also common in both parts.

Thus, by cancelling the 3 and x from both the numerator and the denominator, you significantly reduce the expression's complexity. Recognizing these shared factors is key to simplifying and solving fraction problems efficiently.

Simplest Form of a Fraction

The simplest form of a fraction refers to a fraction that cannot be reduced any further while maintaining the same value. At this stage, the numerator and the denominator have no common factors other than 1.

This is the ultimate goal when simplifying fractions. It ensures the fraction is as compact and straightforward as possible, making it easier to understand or use in further calculations.

From the worked example \(\frac{30x^2}{51xy}\), after factoring and cancelling common factors, the result is \(\frac{10x}{17y}\). To confirm if this new fraction is in its simplest form, you check the numerator and denominator:

• Number 10 is composed of \(2 \times 5\) and is not divisible by 17.
• Variables such as \(x\) and \(y\) are different and not cancellable.

Since there are no additional common factors, \(\frac{10x}{17y}\) is indeed in its simplest form. Getting to this form is essential for accuracy and elegance in mathematical expressions.