What Is 5 5 Divided By 1 2

What is 5 5/8 Divided by 1 1/2? A Deep Dive into Fraction Division

This article will thoroughly explain how to solve the mathematical problem: 5 5/8 divided by 1 1/2. We'll go beyond just providing the answer, delving into the underlying principles of fraction division, offering various methods for solving the problem, and exploring practical applications. This comprehensive guide will equip you with the skills to confidently tackle similar fraction division problems in the future.

Understanding Fractions and Mixed Numbers

Before we tackle the division problem, let's refresh our understanding of fractions and mixed numbers.

• Fractions: A fraction represents a part of a whole. It's expressed as a ratio of two numbers: the numerator (top number) and the denominator (bottom number). For example, in the fraction 3/4, 3 is the numerator and 4 is the denominator.

• Mixed Numbers: A mixed number combines a whole number and a fraction. For example, 5 5/8 represents 5 whole units and an additional 5/8 of a unit.

Converting Mixed Numbers to Improper Fractions

To divide mixed numbers, it's essential to convert them into improper fractions. An improper fraction has a numerator larger than or equal to its denominator. The conversion process involves:

1. Multiply: Multiply the whole number by the denominator.
2. Add: Add the result to the numerator.
3. Keep the Denominator: Retain the original denominator.

Let's convert 5 5/8 to an improper fraction:

1. Multiply: 5 * 8 = 40
2. Add: 40 + 5 = 45
3. Keep the Denominator: The denominator remains 8.

Therefore, 5 5/8 is equivalent to 45/8.

Similarly, let's convert 1 1/2 to an improper fraction:

1. Multiply: 1 * 2 = 2
2. Add: 2 + 1 = 3
3. Keep the Denominator: The denominator remains 2.

Therefore, 1 1/2 is equivalent to 3/2.

Dividing Fractions: The Reciprocal Method

Dividing fractions involves a simple yet powerful technique: multiplying by the reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

For example, the reciprocal of 3/2 is 2/3.

To divide fractions, we follow these steps:

1. Convert to Improper Fractions (if necessary): As demonstrated above.
2. Change Division to Multiplication: Replace the division sign (÷) with a multiplication sign (×).
3. Invert the Second Fraction: Replace the second fraction with its reciprocal.
4. Multiply Numerators and Denominators: Multiply the numerators together and the denominators together.
5. Simplify (if possible): Reduce the resulting fraction to its simplest form.

Solving 5 5/8 Divided by 1 1/2

Now, let's apply these steps to solve 5 5/8 ÷ 1 1/2:

1. Convert to Improper Fractions: We already did this: 45/8 ÷ 3/2

2. Change Division to Multiplication: 45/8 × 3/2

3. Invert the Second Fraction: 45/8 × 2/3

4. Multiply Numerators and Denominators: (45 × 2) / (8 × 3) = 90/24

5. Simplify: We can simplify 90/24 by dividing both the numerator and denominator by their greatest common divisor, which is 6. 90 ÷ 6 = 15 and 24 ÷ 6 = 4.

Therefore, 5 5/8 ÷ 1 1/2 = 15/4

Converting back to a Mixed Number (Optional)

While 15/4 is a perfectly acceptable answer, we can convert it back to a mixed number if desired. To do this:

1. Divide the Numerator by the Denominator: 15 ÷ 4 = 3 with a remainder of 3.
2. The Whole Number: The quotient (3) becomes the whole number part.
3. The Fraction: The remainder (3) becomes the numerator, and the original denominator (4) remains the same.

Therefore, 15/4 is equivalent to 3 3/4.

Alternative Methods: Using Decimals

While the reciprocal method is generally preferred for fraction division, we can also solve this problem using decimal equivalents.

1. Convert Fractions to Decimals: 5 5/8 = 5.625 and 1 1/2 = 1.5
2. Perform Decimal Division: 5.625 ÷ 1.5 ≈ 3.75

Notice that 3.75 is the decimal equivalent of 3 3/4. This confirms our previous result.

Practical Applications of Fraction Division

Fraction division is not just an abstract mathematical concept; it has numerous practical applications in everyday life and various professions. Here are a few examples:

• Cooking and Baking: Scaling recipes up or down requires dividing fractional quantities. For instance, if a recipe calls for 1 1/2 cups of flour and you want to make half the recipe, you'd divide 1 1/2 by 2.

• Sewing and Tailoring: Cutting fabric accurately for garments often involves fraction division. Determining the number of pieces you can cut from a given length of fabric might require dividing fractions.

• Construction and Engineering: Calculating material quantities for projects frequently involves working with fractions and mixed numbers. Dividing fractions helps determine how many pieces of a certain size can be cut from a larger piece.

• Data Analysis and Statistics: Dividing fractions plays a crucial role in various statistical calculations, including calculating proportions, ratios, and averages.

• Financial Calculations: Dividing fractions can be useful in calculating shares, percentages, and other financial proportions.

Conclusion: Mastering Fraction Division

Understanding and mastering fraction division is a fundamental skill with broad applications. This article has provided a comprehensive explanation of how to solve the problem 5 5/8 divided by 1 1/2, highlighting the importance of converting to improper fractions, using the reciprocal method, and simplifying the result. We've also explored alternative approaches using decimals and discussed practical applications in everyday life. By understanding these concepts, you'll be well-equipped to tackle more complex fraction problems and confidently apply this knowledge in various contexts. Remember to practice regularly to solidify your understanding and build your mathematical fluency. The more you practice, the easier it will become!