10/13 Divided By 4/9 In Simplest Form

10/13 Divided by 4/9 in Simplest Form: A Comprehensive Guide

Dividing fractions can seem daunting, but with a structured approach, it becomes a straightforward process. This comprehensive guide will walk you through dividing 10/13 by 4/9, explaining each step in detail, providing alternative methods, and exploring related concepts to build a solid understanding of fraction division. We'll also delve into the importance of simplifying fractions and how to achieve the simplest form consistently.

Understanding Fraction Division

Before tackling the specific problem, let's establish a foundational understanding of dividing fractions. The core principle is to invert the second fraction (the divisor) and multiply. This is often remembered with the phrase "Keep, Change, Flip".

• Keep: Keep the first fraction as it is.
• Change: Change the division sign to a multiplication sign.
• Flip: Flip (invert) the second fraction.

This seemingly simple rule is based on the mathematical properties of reciprocals and multiplication. Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply the fraction flipped upside down.

Step-by-Step Solution: 10/13 ÷ 4/9

Now, let's apply this principle to solve 10/13 ÷ 4/9:

1. Keep: Keep the first fraction: 10/13

2. Change: Change the division sign (÷) to a multiplication sign (×): 10/13 ×

3. Flip: Flip the second fraction (4/9) to its reciprocal (9/4): 10/13 × 9/4

4. Multiply the numerators: Multiply the numerators (top numbers) together: 10 × 9 = 90

5. Multiply the denominators: Multiply the denominators (bottom numbers) together: 13 × 4 = 52

6. Result: This gives us the initial result: 90/52

Simplifying the Fraction

The fraction 90/52 is not in its simplest form. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator (90) and the denominator (52). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the GCD can be done using several methods:

• Listing Factors: List all the factors of 90 and 52 and identify the largest common factor. This method is suitable for smaller numbers.
• Prime Factorization: Break down both numbers into their prime factors. The GCD is the product of the common prime factors raised to the lowest power. This method is more efficient for larger numbers.
• Euclidean Algorithm: This is an efficient algorithm for finding the GCD, especially for larger numbers.

Let's use prime factorization to find the GCD of 90 and 52:

• Prime factorization of 90: 2 × 3² × 5
• Prime factorization of 52: 2² × 13

The only common prime factor is 2, raised to the power of 1 (the lowest power). Therefore, the GCD of 90 and 52 is 2.

Now, divide both the numerator and the denominator by the GCD (2):

• 90 ÷ 2 = 45
• 52 ÷ 2 = 26

This gives us the simplified fraction: 45/26

The Final Answer: 45/26

Therefore, 10/13 divided by 4/9, simplified to its lowest terms, is 45/26. This is an improper fraction (the numerator is larger than the denominator), which can also be expressed as a mixed number. To convert it to a mixed number, divide the numerator (45) by the denominator (26):

45 ÷ 26 = 1 with a remainder of 19.

So, the mixed number representation is 1 19/26.

Alternative Methods

While the "Keep, Change, Flip" method is the most common, there are alternative approaches to dividing fractions:

• Converting to Decimal: Convert both fractions to decimals, then perform the division. This method is useful when using a calculator, but it can lead to rounding errors and may not always provide the simplest fractional form.

• Using Complex Fractions: Represent the division as a complex fraction, where the numerator and denominator are themselves fractions. Then, simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

Importance of Simplifying Fractions

Simplifying fractions is crucial for several reasons:

• Clarity: A simplified fraction is easier to understand and interpret than a complex one.
• Accuracy: In calculations involving fractions, using simplified fractions reduces the risk of errors.
• Efficiency: Simplified fractions are more efficient to use in further calculations.
• Standard Form: Presenting answers in their simplest form is a standard mathematical practice.

Practice Problems

To solidify your understanding, try these practice problems:

1. 5/7 ÷ 2/3
2. 12/15 ÷ 6/10
3. 21/28 ÷ 3/4

Conclusion

Dividing fractions, while initially appearing complex, becomes manageable with a systematic approach. Remembering the "Keep, Change, Flip" method and mastering fraction simplification are key to solving fraction division problems accurately and efficiently. The process involves inverting the second fraction, multiplying, and simplifying the resulting fraction to its simplest form, often using techniques such as prime factorization to find the greatest common divisor. By understanding these steps and practicing regularly, you can confidently tackle various fraction division problems. The ability to work with fractions is fundamental to many areas of mathematics and beyond, making it a skill well worth mastering.