Rewrite Each Fraction Using The Given Denominator

Rewriting Fractions: Mastering Equivalent Fractions with Different Denominators

This comprehensive guide delves into the fundamental concept of rewriting fractions with a specified denominator. We'll explore the core principles, step-by-step methods, and practical applications to solidify your understanding of equivalent fractions. Whether you're a student grappling with fractions or an adult looking to refresh your math skills, this guide will equip you with the tools to confidently tackle any fraction rewriting problem.

Understanding Equivalent Fractions

Before diving into the mechanics of rewriting fractions, let's establish a solid foundation. Equivalent fractions represent the same value, even though they look different. Think of it like slicing a pizza: a pizza cut into 8 slices with 2 slices taken is the same as a pizza cut into 4 slices with 1 slice taken. Both represent one-quarter (1/4) of the pizza.

Key Principle: To create an equivalent fraction, you must multiply or divide both the numerator (top number) and the denominator (bottom number) by the same non-zero number. This crucial step ensures that the value of the fraction remains unchanged.

Visualizing Equivalent Fractions

Imagine a simple fraction like 1/2. To visualize equivalent fractions, consider a rectangular shape divided into two equal parts. Shading one part represents 1/2. Now, imagine dividing each of these two parts in half. You now have four equal parts, and two of them are shaded. This visually demonstrates that 1/2 is equivalent to 2/4.

Continuing this pattern, dividing each of the four parts in half again shows that 1/2 is also equivalent to 4/8, 8/16, and so on. This visual representation clearly illustrates the concept of equivalent fractions and how they maintain the same value despite different numerators and denominators.

Rewriting Fractions with a Given Denominator: Step-by-Step Guide

Rewriting a fraction with a specific denominator involves finding the equivalent fraction that has the desired denominator. Here's a systematic approach:

Step 1: Identify the Target Denominator:

Clearly identify the new denominator you want for your fraction. This is the denominator you aim to achieve in your equivalent fraction.

Step 2: Determine the Multiplication Factor:

Divide the target denominator by the original denominator. The result is the multiplication factor you need to apply to both the numerator and the denominator.

Step 3: Multiply Both the Numerator and Denominator:

Multiply both the numerator and the denominator of the original fraction by the multiplication factor you calculated in Step 2. This will transform your original fraction into an equivalent fraction with the desired denominator.

Step 4: Simplify (if necessary):

After multiplying, check if the resulting fraction can be simplified. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD) to express the fraction in its simplest form. This step is crucial for ensuring the fraction is presented in its most concise and efficient form.

Examples: Rewriting Fractions with Different Denominators

Let's work through some examples to solidify our understanding:

Example 1: Rewrite 2/3 with a denominator of 9.

1. Target Denominator: 9
2. Multiplication Factor: 9 / 3 = 3
3. Multiply: (2 x 3) / (3 x 3) = 6/9
4. Simplify: The fraction 6/9 is already in its simplest form.

Therefore, 2/3 is equivalent to 6/9.

Example 2: Rewrite 3/4 with a denominator of 12.

1. Target Denominator: 12
2. Multiplication Factor: 12 / 4 = 3
3. Multiply: (3 x 3) / (4 x 3) = 9/12
4. Simplify: The fraction 9/12 can be simplified by dividing both numerator and denominator by 3, resulting in 3/4. Note that this confirms our process; the equivalent fraction remains the same value in simplified form.

Thus, 3/4 is equivalent to 9/12.

Example 3: Rewrite 5/6 with a denominator of 18.

1. Target Denominator: 18
2. Multiplication Factor: 18 / 6 = 3
3. Multiply: (5 x 3) / (6 x 3) = 15/18
4. Simplify: The fraction 15/18 can be simplified by dividing both numerator and denominator by 3, resulting in 5/6. Again, the simplification confirms our process.

Therefore, 5/6 is equivalent to 15/18.

Example 4: Dealing with Larger Numbers

Rewrite 7/15 with a denominator of 45

1. Target Denominator: 45
2. Multiplication Factor: 45 / 15 = 3
3. Multiply: (7 x 3) / (15 x 3) = 21/45
4. Simplify: 21/45 does not simplify further because their Greatest Common Factor (GCF) is 3.

Therefore, 7/15 is equivalent to 21/45

Example 5: When the target denominator is smaller than the original

This scenario involves division instead of multiplication. Let's say we want to rewrite 12/18 with a denominator of 6.

1. Target Denominator: 6
2. Division Factor: 6 / 18 = 1/3 (or 0.333...)
3. Divide: (12 ÷ 3) / (18 ÷ 3) = 4/6
4. Simplify: 4/6 can be further simplified to 2/3

Therefore, 12/18 is equivalent to 4/6, which simplifies to 2/3. Note that you must be able to divide both the numerator and denominator by the same whole number to successfully simplify and find an equivalent fraction.

Practical Applications of Rewriting Fractions

The ability to rewrite fractions with a given denominator is crucial in many mathematical contexts:

• Adding and Subtracting Fractions: Before adding or subtracting fractions, they must have a common denominator. Rewriting fractions allows you to find a common denominator efficiently, enabling seamless addition or subtraction.

• Comparing Fractions: Determining which fraction is larger or smaller is easier when both fractions have the same denominator. Rewriting fractions with a common denominator facilitates clear comparison.

• Solving Equations: Many algebraic equations involve fractions. Rewriting fractions to have a common denominator simplifies the equation, making it easier to solve.

• Real-world Problems: From baking recipes that require fractional measurements to calculating portions of a budget, the ability to rewrite fractions is essential for practical problem-solving in everyday life.

Troubleshooting Common Mistakes

• Incorrect Multiplication Factor: The most frequent mistake is calculating the incorrect multiplication factor. Always double-check your division to ensure accuracy.

• Forgetting to Multiply Both Numerator and Denominator: Remember to multiply both the numerator and the denominator by the same number. Failure to do so will alter the value of the fraction.

• Improper Simplification: After multiplying, always simplify the resulting fraction to its lowest terms. This ensures the fraction is expressed in its most efficient form.

Conclusion: Mastering Fraction Rewriting

Rewriting fractions with a given denominator is a fundamental skill in mathematics. Mastering this skill provides a strong foundation for more advanced mathematical concepts and practical problem-solving. By following the step-by-step guide and practicing with various examples, you can build confidence and accuracy in your ability to work with fractions effectively. Remember the key principle: multiplying or dividing both the numerator and denominator by the same non-zero number maintains the fraction's value while changing its appearance. This process is essential for many mathematical operations and real-world applications. Consistent practice will make you proficient in rewriting fractions, opening up a world of mathematical possibilities.