6/35 Divided By 9/7 In Simplest Form

6/35 Divided by 9/7 in Simplest Form: A Comprehensive Guide

Dividing fractions can seem daunting at first, but with a clear understanding of the process and a systematic approach, it becomes straightforward. This comprehensive guide will walk you through the steps of dividing 6/35 by 9/7, explaining each stage in detail and providing valuable insights into fraction manipulation. We'll not only find the solution but also explore the underlying principles, ensuring you can confidently tackle similar problems in the future.

Understanding Fraction Division

Before diving into the specific problem, let's review the fundamental concept of dividing fractions. The core principle is to invert the second fraction (the divisor) and multiply. This seemingly simple rule stems from the definition of division as the inverse of multiplication. When we divide by a fraction, we're essentially asking: "How many times does the divisor fraction fit into the dividend fraction?" Inverting and multiplying provides the elegant and efficient solution.

The Reciprocal: The Key to Fraction Division

The reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of 2/3 is 3/2, and the reciprocal of 5/1 (which is just 5) is 1/5. This concept of the reciprocal is crucial for dividing fractions because we replace division with multiplication by the reciprocal.

Step-by-Step Solution: 6/35 ÷ 9/7

Now, let's tackle the problem at hand: 6/35 divided by 9/7.

Step 1: Invert the Divisor

The first step is to find the reciprocal of the second fraction (the divisor), which is 9/7. The reciprocal of 9/7 is 7/9.

Step 2: Change Division to Multiplication

We replace the division sign (÷) with a multiplication sign (×). Our problem now becomes:

6/35 × 7/9

Step 3: Multiply the Numerators and Denominators

Next, we multiply the numerators together (the top numbers) and the denominators together (the bottom numbers):

(6 × 7) / (35 × 9) = 42/315

Step 4: Simplify the Fraction

The fraction 42/315 is not in its simplest form. To simplify, we need to find the greatest common divisor (GCD) of both the numerator (42) and the denominator (315). The GCD is the largest number that divides both numbers without leaving a remainder.

Finding the GCD can be done through various methods, including prime factorization.

• Prime Factorization of 42: 2 × 3 × 7
• Prime Factorization of 315: 3 × 3 × 5 × 7

The common prime factors are 3 and 7. Therefore, the GCD is 3 × 7 = 21.

Step 5: Divide Both Numerator and Denominator by the GCD

Divide both the numerator and the denominator by the GCD (21):

42 ÷ 21 = 2 315 ÷ 21 = 15

This gives us the simplified fraction:

2/15

Therefore, 6/35 divided by 9/7, in its simplest form, is 2/15.

Alternative Methods and Further Exploration

While the method above is the standard approach, let's explore some alternative methods and delve deeper into the underlying concepts.

Simplifying Before Multiplication

One powerful technique is to simplify the fractions before multiplying. This often makes the calculations significantly easier. Notice that in our original problem, we had:

6/35 × 7/9

Observe that we can cancel common factors between the numerator and denominator before multiplying. The number 7 appears in the numerator (7) and the denominator (35). We can simplify:

7 ÷ 7 = 1 35 ÷ 7 = 5

Our expression now becomes:

6/5 × 1/9

Now we can multiply:

(6 × 1) / (5 × 9) = 6/45

This fraction can be further simplified by dividing both numerator and denominator by their GCD, which is 3:

6 ÷ 3 = 2 45 ÷ 3 = 15

Again, we arrive at the simplified answer: 2/15

This approach demonstrates the efficiency of simplifying before multiplying, reducing the size of the numbers involved and making the calculations less complex.

Visualizing Fraction Division

Imagine you have a pizza cut into 35 slices. You have 6 of these slices (6/35). Now, you want to divide this portion into groups, each containing 9/7 of a pizza (if you had a whole pizza cut into 7 slices, you would take 9 slices from a second pizza). How many of these 9/7 pizza portions can you create from your 6/35 portion? The answer, as we've calculated, is 2/15. This visual representation helps solidify the concept of fraction division.

Applying to Real-World Scenarios

Fraction division frequently arises in various real-world scenarios. Consider these examples:

• Baking: If a recipe calls for 6/35 of a cup of sugar and you want to reduce the recipe to 9/7 of its original size, how much sugar do you need? The answer would be (6/35) ÷ (9/7) = 2/15 of a cup.

• Sewing: If you have 6/35 yards of fabric and need to cut pieces that are each 9/7 yards long, how many pieces can you cut? The answer would again be (6/35) ÷ (9/7) = 2/15. This is a very small fraction so the situation would call for an adjustment in the context.

• Construction: Imagine you have 6/35 of a container of cement mix and each part of your project uses 9/7 of a container. How much of the project can you do? The answer, again, is 2/15.

These examples demonstrate how fraction division is not just an abstract mathematical concept but a practical tool for solving real-world problems.

Conclusion

Dividing fractions, though initially appearing complex, is a straightforward process once you master the technique of inverting the divisor and multiplying. Simplifying before multiplication can significantly streamline the calculation, making it less prone to errors. This guide has provided a thorough understanding of the steps involved in dividing 6/35 by 9/7, arriving at the simplified answer of 2/15, and explored the broader application of this fundamental mathematical concept in various real-world scenarios. Remember, practice is key to mastering this skill, so continue practicing different fraction division problems to build your confidence and proficiency.