Problem 41 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{96 x^{2} y}{108 x y^{2}}$$

Step-by-Step Solution

Answer

The fraction simplifies to \(\frac{8x}{9y}\).

Step 1: Simplify the Fraction

Start by writing the fraction and identify the common factors in the numerator and the denominator. The given fraction is \(\frac{96x^{2}y}{108xy^{2}}\). Observe that both 96 and 108 have a common factor, and similarly, \(x\) and \(y\) are both present in the numerator and the denominator.

Step 2: Find the Greatest Common Divisor (GCD)

The GCD of 96 and 108 is determined by the prime factorization method. You will find that \(96 = 2^5 \times 3^1\) and \(108 = 2^2 \times 3^3\). Hence, the GCD is \(2^2 \times 3^1 = 12\).

Step 3: Divide Coefficients by GCD

Divide both the numerator and the denominator by the GCD of 12. So, \(\frac{96}{12} = 8\) and \(\frac{108}{12} = 9\). This simplifies the fraction to \(\frac{8x^{2}y}{9xy^{2}}\).

Step 4: Simplify Variable Terms

Reduce the powers of \(x\) and \(y\) by canceling out common terms. Since there is \(x^1\) in the denominator, cancel \(x^1\) from both numerator and denominator reducing \(x^{2}\) to \(x\). Similarly, \(y^1\) in the numerator and \(y^2\) in the denominator reduces to \(y\) in the denominator. The fraction now becomes \(\frac{8x}{9y}\).

Step 5: Conclude with the Simplified Fraction

Verify if further simplification is possible. In this case, no further simplification is required as there are no common factors between 8, 9, \(x\), and \(y\). Therefore, the fraction in the lowest terms is \(\frac{8x}{9y}\).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is the largest number that divides two or more numbers without leaving a remainder. It is a crucial concept when simplifying fractions. To find the GCD, a common practice is to use prime factorization. This method involves breaking down each number into its prime factors.

In the fraction \(\frac{96x^2y}{108xy^2}\), the numbers 96 and 108 were decomposed into their prime factors:

• \(96 = 2^5 \times 3^1\)
• \(108 = 2^2 \times 3^3\)

To find the GCD, we select the lowest power of all common prime factors. Here, both numbers share the primes 2 and 3:

• Lowest power of 2: \(2^2\)
• Lowest power of 3: \(3^1\)

Therefore, the GCD is \(2^2 \times 3^1 = 12\). Knowing the GCD allows you to divide both the numerator and the denominator for simplification.

Prime Factorization

Prime factorization is the process of expressing a number as a product of prime numbers. Primes are numbers greater than 1 that have no divisors other than 1 and themselves. Prime factorization is commonly used to find the GCD, as it breaks down numbers to their most basic building blocks.

For example, when working with the fraction \(\frac{96x^2y}{108xy^2}\), the numbers 96 and 108 were each transformed into products of prime numbers:

• 96 became \(2^5 \times 3^1\)
• 108 became \(2^2 \times 3^3\)

By factorizing numbers into their primes, it becomes easier to identify common factors. These factors can be used to simplify fractions or solve a variety of mathematical problems efficiently.

Prime factorization is not only essential for finding GCD but also helpful in a myriad of other mathematical simplification tasks.

Variable Reduction

Variable reduction is the simplification step where variables with exponents are reduced by eliminating common terms in both the numerator and the denominator. This step streamlines expressions into their simplest form.

In our exercise, we see the fraction \(\frac{96x^2y}{108xy^2}\) containing variables \(x\) and \(y\). Upon identifying \(x^1\) in the denominator, we simplify \(x^2\) in the numerator to \(x\) by dividing through by \(x\). Similarly, \(y^1\) in the numerator diminishes \(y^2\) in the denominator to \(y\) effectively giving:

• This results in the fraction transforming to \(\frac{8x}{9y}\).

Variable reduction is crucial because it allows us to see and understand the relationships and ratios clearly without excessive terms. By reducing these terms, it not only simplifies calculations but also enhances clarity.

Mathematical Simplification

Mathematical simplification is about making an expression or fraction as simple as possible. The goal is to eliminate any unnecessary complexity. This involves using basic operations and properties of numbers and variables.

For the fraction \(\frac{96x^2y}{108xy^2}\), simplification involved several steps:

• Identifying and using the GCD to divide numerical coefficients, resulting in \(\frac{8x^2y}{9xy^2}\).
• Utilizing variable reduction to further reduce 'fluff' in the variables, arriving at \(\frac{8x}{9y}\).

Mathematical simplification often culminates in cleaner results. These results are easier to interpret and work with. Simplification ensures that expressions are presented in the most straightforward manner possible. This enables better understanding and effective solutions to mathematical problems.