Problem 97 - Algebra 1: Concepts and Skills

Question

Write three equivalent fractions for the given fraction. $$ \frac{3}{5} $$

Step-by-Step Solution

Answer

The three equivalent fractions to \( \frac{3}{5} \) are \( \frac{6}{10} \), \( \frac{9}{15} \), and \( \frac{12}{20} \)

Step 1: Identify the Given Fraction

The given fraction is \( \frac{3}{5} \). We will multiply both the numerator and the denominator by the same number to get equivalent fractions.

Step 2: Create the First Equivalent Fraction

Multiply both the numerator and denominator of the given fraction by 2. This gives us \( \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \). So, \( \frac{6}{10} \) is equivalent to \( \frac{3}{5} \).

Step 3: Create the Second Equivalent Fraction

This time, multiply both the numerator and denominator of the given fraction by 3. This gives us \( \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \). So, \( \frac{9}{15} \) is equivalent to \( \frac{3}{5} \).

Step 4: Create the Third Equivalent Fraction

Finally, multiply both the numerator and denominator of the given fraction by 4. This gives us \( \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \). So, \( \frac{12}{20} \) is equivalent to \( \frac{3}{5} \).

Key Concepts

Fractions

A fraction represents a part of a whole or, more generally, any number of equal parts. When we write a fraction, we use two numbers separated by a slash. For example, in the fraction \( \frac{3}{5} \), the number 3 is the numerator, representing how many parts we have, and the number 5 is the denominator, indicating the total number of equal parts the whole is divided into.

Understanding fractions is crucial because they are used in daily life, such as in cooking, measuring distances, and in various mathematical concepts. When working with fractions, the most important thing is to remember that they must represent the same size of parts. This leads us to the concept of equivalent fractions, where two or more fractions represent the same portion of a whole even though they may look different.

Creating equivalent fractions can be simply done by multiplying or dividing both the numerator and the denominator by the same non-zero number. This maintains the value of the fraction because you're effectively multiplying it by one.

Numerator and Denominator

The numerator and denominator are the two parts that make up a fraction. The numerator, located above the fraction bar, counts how many parts are being considered, while the denominator, found below the fraction bar, represents the total number of parts that make up a whole.

When discussing equivalent fractions, it's these two components that we manipulate. By consistently altering the numerator and denominator, we retain the fraction's essence. It is essential to understand that changing either the numerator or denominator independently changes the fraction's value. For instance, \( \frac{3}{5} \) and \( \frac{6}{10} \) are equivalent because the numerator and denominator of the second fraction have been multiplied by the same number, 2. This keeps the fractions' values equal even though their appearances are different.

Multiplying Fractions

When creating equivalent fractions, we often multiply both the numerator and the denominator by the same number. This is a fundamental aspect of understanding how to handle fractions mathematically. Multiplying fractions is straightforward as it involves multiplying the numerators together to get the new numerator, and the denominators together to get the new denominator.

For example, when we multiply the numerator and denominator of \( \frac{3}{5} \) by 2, we arrive at the fraction \( \frac{6}{10} \) as shown in the provided solution. Each step of multiplying by 2, 3, or any other number, must be performed to both numerator and denominator to maintain the equivalent value of the fraction.

However, we should always multiply by whole numbers to keep the fractions meaningful. If we were to multiply by a fraction or a decimal, the original fraction's value might no longer be preserved, leading to an incorrect equivalent fraction.