Problem 100 - Algebra 1: Concepts and Skills

Question

Write three equivalent fractions for the given fraction. $$ \frac{2}{3} $$

Step-by-Step Solution

Answer

The three equivalent fractions for \(\frac{2}{3}\) are \(\frac{4}{6}, \frac{6}{9}, \frac{8}{12}\)

Step 1: Multiply by 2

Multiply the numerator and the denominator of the fraction \( \frac{2}{3} \) by 2 to obtain the first equivalent fraction. \[ \frac{2 * 2}{3 * 2} = \frac{4}{6} \]

Step 2: Multiply by 3

Multiply the numerator and the denominator of the initial fraction by 3 to derive the second equivalent fraction. \[ \frac{2 * 3}{3 * 3} = \frac{6}{9} \]

Step 3: Multiply by 4

Multiply the numerator and the denominator of the initial fraction by 4 to get the third equivalent fraction. \[ \frac{2 * 4}{3 * 4} = \frac{8}{12} \]

Key Concepts

Multiplying Fractions

To start understanding fractions better, we must dive into the concept of multiplying fractions. Multiplying fractions involves taking two fractions and multiplying their numerators together and their denominators together. This concept allows us to find equivalent fractions easily.

When you're given a fraction, such as \( \frac{2}{3} \), and need to find other fractions that look different but represent the same value, you multiply both the numerator (top number) and the denominator (bottom number) by the same non-zero number. This process does not change the overall value of the fraction, just its form.

For instance:

• Multiply by 2: \( \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \)
• Multiply by 3: \( \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \)
• Multiply by 4: \( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)

Each of these fractions is equivalent to \( \frac{2}{3} \).
This method keeps the balance of the fraction while giving it a new look.

Numerator and Denominator

A fraction might look like just a set of numbers with a line between them, but each part of a fraction has a specific role. Understanding the terms 'numerator' and 'denominator' is key to mastering fractions.

The numerator is the top part of the fraction. It tells us how many parts of the whole we are considering. In the example \( \frac{2}{3} \), the number 2 is the numerator, signifying two parts of the total that we are dealing with.

The denominator, located at the bottom, informs us of the total number of equal parts making up the whole. Here, 3 is the denominator, indicating the whole is split into three equal parts.

• This differentiation helps us easily manipulate fractions by multiplication, while ensuring the fractions are equivalent when both parts are multiplied equally. For instance, by multiplying the numerator and the denominator by 3, \( \frac{2}{3} \) becomes \( \frac{6}{9} \).
• It's crucial to note that the denominator can never be zero because you cannot divide by zero.

Remembering this will make working with fractions a breeze.

Rational Numbers

The world of numbers is vast, but rational numbers form a significant part of it, encompassing many of the numbers we use. A rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero.

Fractions, such as \( \frac{2}{3} \), are perfect examples of rational numbers. They represent the ratio of two whole numbers. By altering the numerator and denominator equitably, we can express an infinite number of equivalent rational numbers like \( \frac{4}{6} \), \( \frac{6}{9} \), and \( \frac{8}{12} \).

This extends beyond simple fractions. Any integers, such as 5, can be rethought of as a rational number by expressing it as a fraction, like \( \frac{5}{1} \). Understandably:

• They can be either positive, negative, or zero.
• They provide a way to accurately represent portions of a whole.

Comprehending rational numbers as ratios broadens our number sense and enriches our appreciation of mathematical relationships.