Hello there! I'm an educator with a passion for mathematics and a knack for simplifying complex concepts. I'm here to help you understand how to figure out equivalent fractions, which is a fundamental skill in mathematics that allows you to compare and manipulate fractions with ease.

Understanding Fractions
Fractions are a way of representing a part of a whole. A fraction is composed of a numerator, which is the top number, and a denominator, which is the bottom number. The numerator tells you how many parts you have, and the denominator tells you into how many parts the whole is divided.

Equivalent Fractions
Equivalent fractions are fractions that have the same value but are expressed differently. For example, 1/2 and 2/4 are equivalent fractions because they both represent half of a whole.

Finding Equivalent Fractions
Now, let's dive into the process of finding equivalent fractions. The key to this process is to understand that the value of a fraction does not change if both the numerator and the denominator are multiplied or divided by the same nonzero whole number.

Procedure to Find Equivalent Fractions

1. Identify the Fraction: Start with a given fraction, let's say \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator.


2. Choose a Multiplier: Decide on a nonzero whole number that you will use to multiply both the numerator and the denominator. This number can be any integer except zero.


3. Apply the Multiplier: Multiply both the numerator and the denominator by the chosen whole number. Let's denote this number as \( c \). So, you will have \( a \times c \) as the new numerator and \( b \times c \) as the new denominator.


4. Form the New Fraction: The new fraction will be \( \frac{a \times c}{b \times c} \). This fraction is equivalent to the original fraction \( \frac{a}{b} \) because the operation applied to both the numerator and the denominator is the same.

Example
Let's consider the fraction \( \frac{1}{3} \) and find an equivalent fraction.

1. Identify the Fraction: \( \frac{1}{3} \)

2. Choose a Multiplier: Let's choose 2 as our multiplier.

3. Apply the Multiplier: Multiply both the numerator and the denominator by 2:
- New Numerator: \( 1 \times 2 = 2 \)
- New Denominator: \( 3 \times 2 = 6 \)

4. Form the New Fraction: The new fraction is \( \frac{2}{6} \), which is equivalent to \( \frac{1}{3} \).

Why Does This Work?
The reason this method works is rooted in the properties of fractions. Multiplying or dividing both the numerator and the denominator by the same number is essentially scaling up or down the fraction without changing its inherent value. It's similar to scaling a map or a blueprint; the proportions remain the same, just the size changes.

Conclusion
Finding equivalent fractions is a straightforward process that involves multiplying or dividing both the numerator and the denominator by the same nonzero whole number. This method can be used to simplify fractions, compare fractions, or to express a fraction in a form that is more convenient for a particular calculation or context.

Now, let's move on to the translation.

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Procedure:To find equivalent fractions, multiply the numerator AND denominator by the same nonzero whole number. This procedure is used to solve example 4. You can multiply the numerator and the denominator of a fraction by any nonzero whole number, as long as you multiply both by the same whole number!read more >>