1 6 Divided By 1 3 As A Fraction

1 6 Divided by 1 3 as a Fraction: A Comprehensive Guide

Dividing mixed numbers can seem daunting, but with a systematic approach, it becomes straightforward. This comprehensive guide will walk you through dividing 1 6 by 1 3 as a fraction, explaining the process step-by-step, exploring the underlying mathematical principles, and addressing frequently asked questions. Understanding this process will empower you to tackle similar problems with confidence. This guide will also cover the concept of reciprocals, crucial for division involving fractions and mixed numbers.

Understanding Mixed Numbers and Improper Fractions

Before diving into the division, let's solidify our understanding of mixed numbers and improper fractions. A mixed number combines a whole number and a fraction, like 1 6 (one and six-tenths). An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, 16/10 is an improper fraction equivalent to 1 6.

Converting between mixed numbers and improper fractions is essential for division. To convert a mixed number to an improper fraction:

1. Multiply the whole number by the denominator.
2. Add the numerator to the result.
3. Keep the same denominator.

Let's convert 1 6 to an improper fraction:

1. 1 (whole number) x 10 (denominator) = 10
2. 10 + 6 (numerator) = 16
3. The improper fraction is 16/10.

Similarly, to convert an improper fraction to a mixed number:

1. Divide the numerator by the denominator.
2. The quotient becomes the whole number.
3. The remainder becomes the new numerator.
4. The denominator stays the same.

Step-by-Step Division: 1 6 ÷ 1 3

Now, let's tackle the division problem: 1 6 ÷ 1 3.

Step 1: Convert Mixed Numbers to Improper Fractions

First, we convert both mixed numbers into improper fractions:

• 1 6 = 16/10
• 1 3 = 13/10

Our problem now becomes: 16/10 ÷ 13/10

Step 2: The Reciprocal Rule

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 13/10 is 10/13.

Step 3: Multiplication

Rewrite the division as a multiplication problem by using the reciprocal of the second fraction:

16/10 ÷ 13/10 = 16/10 * 10/13

Step 4: Simplify and Multiply

Before multiplying, we can simplify by canceling common factors between the numerators and denominators. Notice that both 10 in the numerator and 10 in the denominator cancel each other out:

16/1 * 1/13 = 16/13

Step 5: Convert to a Mixed Number (Optional)

The result 16/13 is an improper fraction. We can convert it to a mixed number:

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

Therefore, 16/13 = 1 3/13

Therefore, 1 6 ÷ 1 3 = 1 3/13

The Mathematical Rationale: Why Does This Work?

The process of inverting and multiplying when dividing fractions stems from the fundamental definition of division. Division asks the question: "How many times does one number fit into another?"

Let's consider a simpler example: 2 ÷ 1/2. This asks, "How many halves are there in two wholes?" The answer is four. If we convert this to multiplication using reciprocals, we get 2 x 2/1 = 4.

The same logic applies to mixed numbers. By converting them to improper fractions, we're expressing the numbers in a form that allows for consistent application of the reciprocal rule, enabling us to seamlessly perform the division.

Practical Applications and Real-World Examples

Understanding division with mixed numbers has numerous real-world applications:

• Cooking and Baking: Scaling recipes up or down requires dividing mixed number quantities of ingredients.
• Sewing and Crafts: Calculating fabric lengths or bead counts involves dividing mixed numbers.
• Construction and Engineering: Dividing measurements accurately is vital for precise calculations.
• Finance: Sharing expenses or calculating proportions of investments often requires dividing mixed numbers.

Frequently Asked Questions (FAQ)

Q: Can I divide mixed numbers without converting to improper fractions?

A: While technically possible with a more complex method involving separate division of whole numbers and fractions, converting to improper fractions offers a significantly more efficient and less error-prone method.

Q: What if the denominators are different?

A: If the denominators of the mixed numbers are different, find a common denominator before converting to improper fractions. This ensures consistent calculations throughout the process.

Q: Is there a way to check my answer?

A: Yes! You can check your answer by multiplying your result by the divisor. If the calculation is correct, you should obtain the dividend. In our example: (1 3/13) x (1 3/10) should approximately equal 1 6. (Note: due to rounding in mixed numbers, there might be minor discrepancies).

Q: What if I get a decimal as a result instead of a fraction?

A: Dividing fractions will sometimes result in a fraction that can be converted to a terminating or recurring decimal. The choice between fractional and decimal representation depends on the context of the problem and required precision.

Conclusion

Dividing mixed numbers might seem intimidating initially, but by systematically following the steps of converting to improper fractions, applying the reciprocal rule for division, simplifying where possible, and converting back to a mixed number if necessary, the process becomes manageable and straightforward. Understanding the underlying principles not only helps you solve these problems but also provides a strong foundation for tackling more advanced mathematical concepts. This process, while focusing on a specific example, applies universally to all mixed number divisions. Practice and a clear understanding of these steps will equip you with the confidence to tackle similar problems with ease. Remember to always check your work using multiplication to ensure accuracy.