Order Of Operations On Fractions

Mastering the Order of Operations with Fractions: A Comprehensive Guide

Fractions can be tricky, and when combined with the order of operations (often remembered by the acronym PEMDAS/BODMAS), they can seem downright daunting. This comprehensive guide will break down the process step-by-step, equipping you with the confidence to tackle any fraction-based calculation. We'll cover the fundamental principles, provide practical examples, and address common misconceptions to solidify your understanding. This article is perfect for students struggling with fraction arithmetic and anyone looking to refresh their mathematical skills.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before diving into fractions, let's refresh our understanding of the order of operations. This dictates the sequence in which we perform calculations within an equation to arrive at the correct answer. The acronyms PEMDAS and BODMAS represent the same order, with slight variations in terminology:

• PEMDAS: Parentheses (Brackets), Exponents (Orders), Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• BODMAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

The key takeaway is that operations within parentheses or brackets are always performed first. Then, exponents or orders (powers and roots) are addressed. Multiplication and division have equal precedence and are done from left to right, followed by addition and subtraction, also from left to right.

Working with Fractions: A Quick Review

To confidently handle the order of operations with fractions, we need a solid grasp of basic fraction arithmetic:

• Adding and Subtracting Fractions: Requires a common denominator. For example, to add 1/2 + 1/4, we convert 1/2 to 2/4, resulting in 2/4 + 1/4 = 3/4.
• Multiplying Fractions: Simply multiply the numerators together and the denominators together. For example, (1/2) * (1/4) = 1/8.
• Dividing Fractions: Invert the second fraction (reciprocal) and multiply. For example, (1/2) ÷ (1/4) = (1/2) * (4/1) = 4/2 = 2.
• Simplifying Fractions: Reduce the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, 4/8 simplifies to 1/2 (GCD is 4).

Applying the Order of Operations to Fractions: Step-by-Step Examples

Let's work through several examples demonstrating the application of PEMDAS/BODMAS to fraction problems:

Example 1:

(1/2 + 1/4) * 2 - 1/3

1. Parentheses/Brackets: First, we solve the expression within the parentheses: 1/2 + 1/4 = 3/4. The equation now becomes (3/4) * 2 - 1/3.

2. Multiplication and Division (from left to right): Next, we perform the multiplication: (3/4) * 2 = 6/4 = 3/2. The equation simplifies to 3/2 - 1/3.

3. Addition and Subtraction (from left to right): Finally, we subtract the fractions, finding a common denominator: 3/2 - 1/3 = (9/6) - (2/6) = 7/6.

Therefore, the answer is 7/6.

Example 2:

1/2 + 1/3 ÷ 1/4 - 1/5

1. Division: We begin with the division operation: 1/3 ÷ 1/4 = 1/3 * 4/1 = 4/3. The equation becomes 1/2 + 4/3 - 1/5.

2. Addition and Subtraction (from left to right): We find a common denominator for all the fractions: 1/2 + 4/3 - 1/5 = (15/30) + (40/30) - (6/30) = (15 + 40 - 6) / 30 = 49/30.

Therefore, the answer is 49/30.

Example 3:

(2/3)^2 + 1/2 * (1/4 - 1/8)

1. Parentheses/Brackets: First, we solve the expression within the inner parentheses: 1/4 - 1/8 = 1/8. The equation becomes (2/3)^2 + 1/2 * (1/8).

2. Exponents/Orders: Next, we handle the exponent: (2/3)^2 = (2/3) * (2/3) = 4/9. The equation becomes 4/9 + 1/2 * (1/8).

3. Multiplication: Now we perform the multiplication: 1/2 * 1/8 = 1/16. The equation is simplified to 4/9 + 1/16.

4. Addition: We find a common denominator for the addition: 4/9 + 1/16 = (64/144) + (9/144) = 73/144.

Therefore, the answer is 73/144.

Example 4: Involving Mixed Numbers

2 1/2 ÷ (1 1/3 + 1/6)

1. Parentheses/Brackets: We convert the mixed numbers to improper fractions and solve the addition within the parentheses: 1 1/3 + 1/6 = (4/3) + (1/6) = (8/6) + (1/6) = 9/6 = 3/2. The equation becomes 5/2 ÷ 3/2.

2. Division: We perform the division operation: 5/2 ÷ 3/2 = 5/2 * 2/3 = 10/6 = 5/3.

Therefore, the answer is 5/3 or 1 2/3.

Addressing Common Mistakes

Several common pitfalls can lead to incorrect answers when working with fractions and the order of operations:

• Ignoring Parentheses/Brackets: Always perform operations within parentheses or brackets first. Ignoring this rule will almost always lead to an incorrect result.
• Incorrect Order of Multiplication and Division: Remember to perform multiplication and division from left to right, not necessarily multiplication first.
• Forgetting Common Denominators: Incorrectly adding or subtracting fractions without finding a common denominator is a frequent error.
• Improper Handling of Mixed Numbers: Ensure you correctly convert mixed numbers to improper fractions before performing operations, especially when multiplication or division is involved.

Scientific Notation with Fractions

While less common in basic fraction problems, scientific notation can be applied to very large or very small fractions. You follow the same order of operations, but you deal with the scientific notation before applying the operations. For example, (2 x 10^-3) + (5 x 10^-4) requires adjusting the exponents to have a common base before addition. In this case, you could convert 2 x 10^-3 to 20 x 10^-4 and then add.

Frequently Asked Questions (FAQs)

Q: What if I have nested parentheses (parentheses within parentheses)?

A: Work from the innermost set of parentheses outwards. Solve the expression inside the innermost parentheses first, then the next set, and so on, following PEMDAS/BODMAS within each set.

Q: Can I use a calculator to solve these problems?

A: Yes, most scientific calculators correctly handle the order of operations and fraction calculations. However, understanding the underlying principles is crucial for error checking and problem-solving when calculators are unavailable.

Q: How can I improve my skills in working with fractions and the order of operations?

A: Practice is key! Work through numerous examples, starting with simpler problems and gradually increasing complexity. Use online resources, textbooks, or work with a tutor for additional support and guidance.

Conclusion

Mastering the order of operations with fractions is achievable with consistent practice and a clear understanding of the fundamental principles. By meticulously following PEMDAS/BODMAS and mastering basic fraction arithmetic, you'll gain the confidence to tackle any fraction-based calculation with accuracy and efficiency. Remember to break down complex problems into smaller, manageable steps, and always double-check your work! With dedicated effort, you'll become proficient in this essential mathematical skill.