Problem 77 - Prealgebra

Question

Write each fraction as an equivalent fraction with denominator \(10 .\) $$\frac{3}{5}$$

Step-by-Step Solution

Answer

The fraction \( \frac{3}{5} \) as an equivalent fraction with denominator 10 is \( \frac{6}{10} \).

Step 1: Identify the Relationship

To convert the fraction \( \frac{3}{5} \) into an equivalent fraction with a denominator of 10, we need to find the relationship between the current denominator (5) and the desired denominator (10).

Step 2: Determine the Multiplier

Since 10 is double 5, we need to multiply 5 by 2 to reach 10. This means our multiplier is 2.

Step 3: Apply the Multiplier to the Numerator

Since we multiplied the denominator by 2, we must do the same to the numerator to preserve the value of the fraction. Multiply the numerator, 3, by 2: \[3 \times 2 = 6\]

Step 4: Write the Equivalent Fraction

After multiplying both the numerator and denominator by 2, we find our equivalent fraction: \[\frac{3 \times 2}{5 \times 2} = \frac{6}{10}\] Thus, \( \frac{3}{5} \) is equivalent to \( \frac{6}{10} \).

Key Concepts

Fraction Multiplication

Fraction multiplication is a key concept in understanding how to find equivalent fractions. When we multiply fractions, we're essentially scaling both the numerator and denominator by the same factor. This process allows us to create fractions that are equal in value but have different numbers representing that value.

In the exercise at hand, we started with the fraction \(\frac{3}{5}\) and wanted to express it with a new denominator of 10. By recognizing that we needed to double the denominator, multiplying both parts of the fraction by the same number was the solution. This is a straightforward example of fraction multiplication—scale the numerator and denominator by the same number.

• Multiplying the denominator by 2 turns 5 into 10. This adjustment affects the entire fraction.
• To maintain the fraction's value, the numerator must also be multiplied by the same factor.

By applying such manipulations, we ensure that the new fraction \(\frac{6}{10}\) holds the same value as \(\frac{3}{5}\), highlighting the essence of fraction multiplication.

Denominator Conversion

Denominator conversion is about altering the fraction's denominator to a desired number without changing the fraction's value. This exercise invites us to delve into the details of this process.

The denominator is a crucial part of a fraction, indicating how many total parts the whole is divided into. Converting it to a specific number—like converting \(\frac{3}{5}\) to have a denominator of 10—involves finding a commonality between the old and new denominators.

• Identify the needed change: Moving from 5 to 10 required doubling the denominator, a simple multiplication operation.
• Implement the conversion: By multiplying both parts of the fraction by 2, we ensure that the numerator and denominator's ratio stays intact.

This change is vital to practice because it forms the basis for adding, subtracting, and comparing fractions effectively. Mastering denominator conversion helps students become more adept with fractions as a whole.

Fractions Education

Teaching and understanding fractions are cornerstones of math education, laying the groundwork for more advanced math topics. At their core, fractions represent parts of a whole—letting us measure and express quantities that aren't whole numbers. This simple concept unlocks countless applications and deeper understandings of mathematics later on.

When learning about fractions, students should engage with key concepts like equivalent fractions and operations such as addition, subtraction, multiplication, and division. Practical exercises like converting denominators and understanding fraction multiplication are crucial. They not only enhance computational skills but also deepen conceptual comprehension.

• Engage students with hands-on activities: Such exercises improve understanding.
• Emphasize multiplication and division as foundational operations within fractions.
• Highlight real-world fraction applications: Such as cooking or measuring, to help relate abstract numbers to everyday life.

By fostering a thorough command of fractions, educators equip students to tackle more complex math problems with confidence and enthusiasm.