Problem 42 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{66 x^{2} y}{84 x y^{2}}$$

Step-by-Step Solution

Answer

\( \frac{11 x}{14 y} \)

Step 1: Identify the Greatest Common Divisor (GCD) for Coefficients

The given fraction is \( \frac{66 x^{2} y}{84 x y^{2}} \). First, identify the GCD of the coefficients 66 and 84. The prime factorizations are 66 = 2 \times 3 \times 11 and 84 = 2 \times 2 \times 3 \times 7. The common factors are 2 and 3, so the GCD is 2 \times 3 = 6.

Step 2: Divide Coefficients by the GCD

Divide both the numerator and the denominator by their GCD. Thus, we have \( \frac{66 \div 6}{84 \div 6} = \frac{11}{14} \).

Step 3: Simplify the Variables

For the variables, \( x^2 \) in the numerator and \( x \) in the denominator simplify to \( x \). For \( y \) in the numerator and \( y^2 \) in the denominator, they simplify to \( \frac{1}{y} \). Therefore, the variables simplify to \( x \times \frac{1}{y} \).

Step 4: Compose the Simplified Fraction

Combine the simplified coefficients and variables. The fraction reduces to \( \frac{11 x}{14 y} \).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial tool in mathematics, especially when it comes to simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. Understanding how to find the GCD helps simplify algebraic expressions efficiently.

To find the GCD:

• List the prime factors of each number.
• Identify all common factors between the two lists.
• Multiply the common factors together to get the GCD.

In our exercise, the coefficients 66 and 84 were decomposed into their prime factors: 66 as \(2 \times 3 \times 11\), and 84 as \(2 \times 2 \times 3 \times 7\). The common prime factors are 2 and 3, which means the GCD is \(2 \times 3 = 6\). By dividing both the numerator and denominator by the GCD, fractions can be reduced to their simplest form.

Simplifying Fractions

Simplifying fractions is the process of reducing the numerator and the denominator so that they have no common divisors other than 1. This is often done to make working with fractions easier and the results clearer.Here’s how you go about simplifying fractions:

• First, determine the GCD of the numbers.
• Then, divide both the numerator and the denominator by this GCD to reduce the fraction.
• Also consider any common factors in variables within the fraction.

In the example fraction \(\frac{66x^2y}{84xy^2}\), once we determined the GCD of 66 and 84 was 6, we divided each by 6. This simplified the coefficients to \(\frac{11}{14}\).

Next, simplification of variables continued by reducing \(x^2\) and \(x\) to \(x\), and \(y\) and \(y^2\) to \(\frac{1}{y}\). Therefore, your fraction is now \(\frac{11x}{14y}\), free of any further common factors.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks for algebra and are used extensively in mathematics. In the context of fractions, they include variables alongside numbers (coefficients) that are subject to division and simplification. To simplify algebraic expressions:

• Look for the GCD to reduce coefficients.
• Apply the rules of exponents to simplify variable terms. For instance, \(x^2\) divided by \(x\) simplifies to \(x\).
• Cancel out common terms whenever possible.

This technique of simplifying algebraic expressions within fractions not only helps achieve simpler results, like in our fraction where \(x^2\) and \(x\) simplified, but also aids in solving more complex algebraic problems. Remember, simplified algebraic expressions make it easier to understand and solve equations, perform operations, and interpret mathematical relationships.