Problem 32 - Prealgebra and Introductory Algebra

Question

Write the fractions in terms of the LCM of the denominators. $$\frac{5 y}{6 x^{2}}, \frac{7}{9 x y}$$

Step-by-Step Solution

Answer

\(\frac{5y}{6x^{2}}\) is equivalent to \(\frac{15y^{2}}{18x^{2}y}\) and \(\frac{7}{9xy}\) is equivalent to \(\frac{14x}{18x^{2}y}\) when written in terms of the LCM of the denominators.

Step 1: Identify Denominators

The denominators in our two fractions are \(6x^{2}\) and \(9xy\). We need to find the LCM of these two expression.

Step 2: Calculate the LCM

The Lowest Common Multiple (LCM) of the denominators is found by multiplying the highest power of each factor present in any of the denominators. That is, LCM = \(2*3^{2}*x^{2}*y = 18x^{2}y\).

Step 3: Rewrite the Fractions

Now, write equivalent fractions with their denominators as the LCM. For the fraction \(\frac{5y}{6x^{2}}\) we multiply both the numerator and denominator by \(3*y\) which is \(1\), to get the equivalent fraction \(\frac{15y^{2}}{18x^{2}y}\). For the second fraction, \(\frac{7}{9xy}\), we multiply both the numerator and denominator by \(2*x\) to get \(\frac{14x}{18x^{2}y}\).

Key Concepts

Fractions

In mathematics, fractions are a way to represent a part of a whole. A fraction consists of two numbers: a numerator and a denominator. The numerator is the top number and tells you how many parts you have. The denominator is the bottom number and indicates into how many parts the whole is divided.

• An example of a fraction is \( \frac{3}{4} \), where 3 is the numerator, and 4 is the denominator.
• Fractions can represent values less than 1, equal to 1, or greater than 1.

Understanding fractions is crucial when we perform arithmetic operations, such as addition, subtraction, multiplication, and division. Fractions become particularly interesting when we need to combine them, because this often involves finding a common denominator.

Denominators

The denominator of a fraction is pivotal because it shows the number of equal parts into which the whole is divided. To add or subtract fractions successfully, they must have the same denominator, known as a common denominator.
The Lowest Common Multiple (LCM) of the denominators provides the smallest shared base for fractions. Finding the LCM allows us to convert fractions to equivalent fractions with a common denominator.
In our exercise, the denominators are \(6x^{2}\) and \(9xy\). Finding the LCM, which is \(18x^{2}y\), allows us to rewrite fractions precisely and facilitate operations. It involves taking the highest power of each factor present in any denominator, ensuring the fractions are equivalent upon conversion.

Equivalent Fractions

An equivalent fraction is a different fraction that represents the same value or proportion of the whole. We obtain equivalent fractions by multiplying or dividing the numerator and denominator of a fraction by the same non-zero number.
To express fractions like \( \frac{5y}{6x^{2}} \) and \( \frac{7}{9xy} \) with the LCM \(18x^{2}y\) as the denominator, we need to create equivalent fractions.

• For \( \frac{5y}{6x^{2}} \), multiply by \( \frac{3y}{3y} \) to get \( \frac{15y^{2}}{18x^{2}y} \).
• For \( \frac{7}{9xy} \), multiply by \( \frac{2x}{2x} \) to arrive at \( \frac{14x}{18x^{2}y} \).

This transformation ensures that both fractions share the same denominator, facilitating comparison or further operations. Equivalent fractions are key in solving problems involving different denominators.