5/6 Divided By 2/7 As A Fraction

5/6 Divided by 2/7 as a Fraction: A Comprehensive Guide

Dividing fractions can seem daunting at first, but with a clear understanding of the process and a bit of practice, it becomes straightforward. This article will delve into the division of 5/6 by 2/7, providing a step-by-step explanation, exploring the underlying mathematical principles, and offering practical applications and related examples to solidify your understanding. We'll also touch upon the importance of simplifying fractions and how to check your answers for accuracy.

Understanding Fraction Division

Before diving into the specific problem of 5/6 divided by 2/7, let's establish a foundational understanding of fraction division. The core principle is to invert (reciprocate) the second fraction (the divisor) and then multiply the two fractions.

This method stems from the concept that division is the inverse operation of multiplication. When you divide by a fraction, you're essentially asking, "How many times does the second fraction fit into the first fraction?" Inverting and multiplying provides the answer to this question.

Key Concept: Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of 2/7 is 7/2.

Step-by-Step Solution: 5/6 ÷ 2/7

Now, let's tackle the problem at hand: 5/6 divided by 2/7. We'll break down the solution into clear, manageable steps:

Step 1: Invert the Second Fraction (Divisor)

The second fraction, 2/7, becomes its reciprocal: 7/2.

Step 2: Change the Division Sign to a Multiplication Sign

Replace the division symbol (÷) with a multiplication symbol (×). Our problem now looks like this: 5/6 × 7/2.

Step 3: Multiply the Numerators and the Denominators

Multiply the numerators (the top numbers) together: 5 × 7 = 35. Multiply the denominators (the bottom numbers) together: 6 × 2 = 12.

This gives us the result: 35/12.

Step 4: Simplify the Fraction (If Possible)

In this case, 35 and 12 don't share any common factors other than 1, meaning the fraction is already in its simplest form. However, if there were a common factor (a number that divides evenly into both the numerator and the denominator), we would divide both by that factor to simplify. For example, if the result had been 40/16, we could simplify it to 5/2 by dividing both the numerator and the denominator by 8.

The Result: 35/12

Therefore, 5/6 divided by 2/7 equals 35/12. This is an improper fraction (the numerator is larger than the denominator). We can convert this to a mixed number if desired.

Converting to a Mixed Number

An improper fraction can be expressed as a mixed number, which combines a whole number and a proper fraction. To convert 35/12 to a mixed number, we perform the following steps:

Step 1: Divide the Numerator by the Denominator

Divide 35 by 12: 35 ÷ 12 = 2 with a remainder of 11.

Step 2: Write the Result as a Mixed Number

The quotient (2) becomes the whole number, and the remainder (11) becomes the numerator of the fraction. The denominator remains the same (12).

So, 35/12 as a mixed number is 2 11/12.

Practical Applications and Real-World Examples

Understanding fraction division is crucial in various real-world scenarios:

• Cooking and Baking: Scaling recipes up or down requires dividing fractions. If a recipe calls for 2/3 cup of flour and you want to make half the recipe, you'd divide 2/3 by 2.

• Sewing and Crafts: Cutting fabric or other materials often involves working with fractions of inches or centimeters.

• Construction and Engineering: Precise measurements and calculations in construction and engineering projects frequently require working with fractions.

• Finance and Budgeting: Dividing your budget into fractions for different expenses helps in financial planning.

Further Practice and Related Problems

To solidify your understanding, try solving these similar problems:

• 3/4 ÷ 1/2
• 7/8 ÷ 3/4
• 1/3 ÷ 5/6
• 2/5 ÷ 4/15

Remember to follow the same steps: invert the second fraction, multiply, and simplify.

Checking Your Answer

It's always a good idea to check your answer to ensure accuracy. One way to do this is to perform the inverse operation – multiplication. If 5/6 ÷ 2/7 = 35/12, then 35/12 × 2/7 should equal 5/6. Let's verify:

35/12 × 2/7 = (35 × 2) / (12 × 7) = 70/84

Now, simplify 70/84 by dividing both the numerator and denominator by their greatest common divisor, which is 14:

70/84 = 5/6

This confirms our initial calculation: 5/6 ÷ 2/7 = 35/12.

The Importance of Simplifying Fractions

Simplifying fractions is crucial for several reasons:

• Clarity: Simplified fractions are easier to understand and work with.

• Accuracy: Simplifying fractions reduces the risk of errors in further calculations.

• Efficiency: Simplified fractions make calculations more efficient.

Always simplify your fraction to its lowest terms whenever possible.

Conclusion

Dividing fractions might seem intimidating at first, but by understanding the process of inverting and multiplying, and by practicing consistently, you'll master this essential mathematical skill. This comprehensive guide has provided a step-by-step solution to the problem of 5/6 divided by 2/7, along with practical applications and examples to help you solidify your knowledge. Remember to always check your answer to ensure accuracy and simplify your fractions to their lowest terms for clarity and efficiency. With practice, you'll confidently tackle any fraction division problem that comes your way.