5 5/8 Divided By 2 In Fraction

5 5/8 Divided by 2: A Comprehensive Guide to Fraction Division

Dividing fractions can seem daunting, especially when mixed numbers are involved. This comprehensive guide will walk you through the process of solving 5 5/8 divided by 2, explaining each step clearly and providing additional examples to solidify your understanding. We'll explore various methods, including converting mixed numbers to improper fractions and simplifying the results. By the end, you'll be confident in tackling similar fraction division problems.

Understanding the Problem: 5 5/8 ÷ 2

Our problem is to divide the mixed number 5 5/8 by the whole number 2. This means we're essentially finding out how many times 2 fits into 5 5/8. Before we dive into the solution, let's refresh our understanding of key concepts:

Mixed Numbers and Improper Fractions

A mixed number combines a whole number and a fraction (e.g., 5 5/8). An improper fraction has a numerator (top number) that is greater than or equal to its denominator (bottom number) (e.g., 45/8). These two forms are interchangeable, and converting between them is crucial for fraction division.

The Reciprocal

The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is 1/2, and the reciprocal of 3/4 is 4/3. Understanding reciprocals is essential for dividing fractions.

Method 1: Converting to Improper Fractions

This is generally the most straightforward method for dividing mixed numbers. We'll convert both the mixed number and the whole number into improper fractions before performing the division.

Step 1: Convert the Mixed Number to an Improper Fraction

To convert 5 5/8 to an improper fraction:

1. Multiply the whole number (5) by the denominator (8): 5 * 8 = 40
2. Add the numerator (5) to the result: 40 + 5 = 45
3. Keep the same denominator (8): The improper fraction is 45/8.

Step 2: Convert the Whole Number to an Improper Fraction

To convert the whole number 2 to an improper fraction, simply place it over 1: 2/1.

Step 3: Perform the Division

Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction:

(45/8) ÷ (2/1) = (45/8) * (1/2)

Step 4: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together:

(45 * 1) / (8 * 2) = 45/16

Step 5: Simplify the Result (if necessary)

The improper fraction 45/16 can be converted back into a mixed number:

1. Divide the numerator (45) by the denominator (16): 45 ÷ 16 = 2 with a remainder of 13.
2. The whole number part is 2.
3. The remainder (13) becomes the new numerator, and the denominator remains the same (16).

Therefore, 45/16 simplifies to 2 13/16.

Method 2: Dividing Directly (Less Common but Useful)

While less frequently taught, you can divide a mixed number by a whole number directly. This method requires a bit more intuitive understanding of fractions.

Step 1: Divide the Whole Number Part

Divide the whole number part of the mixed number (5) by the whole number divisor (2): 5 ÷ 2 = 2 with a remainder of 1.

Step 2: Handle the Remainder and Fraction

The remainder (1) is combined with the fractional part (5/8) to form a new fraction: 1 5/8. Now, divide this new fraction by 2: (1 5/8) ÷ 2

Step 3: Convert to Improper Fraction and Solve

Convert 1 5/8 to an improper fraction: (1 * 8 + 5) / 8 = 13/8. Then, divide by 2: (13/8) ÷ (2/1) = (13/8) * (1/2) = 13/16.

Step 4: Combine Results

Add the result from step 1 to the result from step 3: 2 + 13/16 = 2 13/16

Further Examples and Practice

Let's solidify our understanding with a few more examples:

Example 1: 3 1/4 ÷ 5

1. Convert 3 1/4 to an improper fraction: (3 * 4 + 1) / 4 = 13/4
2. Convert 5 to an improper fraction: 5/1
3. Divide: (13/4) ÷ (5/1) = (13/4) * (1/5) = 13/20

Example 2: 7 2/3 ÷ 2

1. Convert 7 2/3 to an improper fraction: (7 * 3 + 2) / 3 = 23/3
2. Convert 2 to an improper fraction: 2/1
3. Divide: (23/3) ÷ (2/1) = (23/3) * (1/2) = 23/6 = 3 5/6

Example 3: 4 3/5 ÷ 3

1. Convert 4 3/5 to an improper fraction: (4 * 5 + 3) / 5 = 23/5
2. Convert 3 to an improper fraction: 3/1
3. Divide: (23/5) ÷ (3/1) = (23/5) * (1/3) = 23/15 = 1 8/15

Troubleshooting Common Mistakes

• Forgetting to use the reciprocal: Remember, when dividing fractions, you multiply by the reciprocal of the second fraction. This is a crucial step.
• Incorrect conversion to improper fractions: Double-check your calculations when converting mixed numbers to improper fractions. A small error here will lead to an incorrect final answer.
• Not simplifying the result: Always simplify your final answer to its lowest terms. This makes the answer easier to understand and interpret.

Conclusion

Dividing mixed numbers by whole numbers might seem challenging initially, but by understanding the fundamental concepts of improper fractions and reciprocals, the process becomes much more manageable. Practice makes perfect – work through the examples provided, and try some problems on your own to build confidence and proficiency in solving fraction division problems. Remember to always check your work and simplify your final answer for the clearest and most accurate result. Mastering fraction division is a crucial skill in mathematics, paving the way for success in more advanced mathematical concepts.