Solving $6 / 7 \div 36 / 56$ Step-by-Step With Simplest Form Answer

Hey guys! Let's break down this math problem together and make sure we get the simplest answer. We're diving into dividing fractions and simplifying them, which is a super important skill in mathematics. You'll find this comes in handy in all sorts of situations, from baking to more advanced math down the road. So, let's jump right in and make sure we understand every step!

Understanding the Problem: $6 / 7 \div 36 / 56$

Okay, so the problem we're tackling is $6 / 7 \div 36 / 56$. The key here is understanding that dividing by a fraction is the same as multiplying by its reciprocal. Think of the reciprocal as flipping the fraction over. So, instead of dividing by $36 / 56$, we're going to multiply by $56 / 36$. This is a fundamental rule in fraction division, and it's what makes these types of problems solvable. Remember, this trick works because division is the inverse operation of multiplication. When we multiply by the reciprocal, we're essentially undoing the division. It's like saying, "How many times does $36 / 56$ fit into $6 / 7$?" which is a bit of a mouthful. But when we flip it and multiply, it becomes much clearer.

Now, let’s rewrite our problem: $6 / 7 \div 36 / 56$ becomes $6 / 7 \times 56 / 36$. This change is crucial. We've transformed a division problem into a multiplication problem, which is generally easier to handle. Make sure you're comfortable with this first step, as it’s the foundation for solving the rest of the problem. When you see a division of fractions, your brain should immediately think, "Flip the second fraction and multiply!" This will save you a lot of headaches down the road.

Multiplying the Fractions

Alright, now that we've flipped the second fraction and turned our division problem into multiplication, we have $6 / 7 \times 56 / 36$. The next step is to simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we're doing $6 \times 56$ for the new numerator and $7 \times 36$ for the new denominator. Let’s break this down:

• Numerator: $6 \times 56 = 336$
• Denominator: $7 \times 36 = 252$

This gives us the fraction $336 / 252$. Now, this fraction looks pretty big and intimidating, right? That's because we haven't simplified it yet. But don't worry, we'll get there! The important thing is that we've correctly multiplied the fractions together. We now have a single fraction that represents the answer to our original division problem, but it’s not in its simplest form. Remember, simplifying fractions means finding the smallest possible numbers that represent the same value. It's like saying we want to express the fraction in its most basic form, where the numerator and denominator have no common factors other than 1. This makes the fraction much easier to understand and work with in the future.

Before we move on to simplifying, let's just recap what we've done so far. We changed the division problem into a multiplication problem by flipping the second fraction, and then we multiplied the numerators and the denominators. This gave us the fraction $336 / 252$. Now comes the fun part: simplifying!

Simplifying the Fraction

Okay, we've got $336 / 252$, and it's time to simplify this bad boy. Simplifying fractions is all about finding the greatest common factor (GCF) of the numerator and the denominator and then dividing both by that factor. The GCF is the largest number that divides evenly into both numbers. There are a couple of ways we can find the GCF, but let's start by trying to find common factors by inspection. This means we'll look at the numbers and see if we can spot any obvious factors.

First, notice that both 336 and 252 are even numbers, which means they're both divisible by 2. Let's divide both by 2:

• $336 \div 2 = 168$
• $252 \div 2 = 126$

So, our fraction becomes $168 / 126$. Still pretty big, huh? Let's keep going. We can see that 168 and 126 are also even, so we can divide by 2 again:

• $168 \div 2 = 84$
• $126 \div 2 = 63$

Now we have $84 / 63$. They're not both even anymore, but let's think about other factors. Do you notice anything special about 84 and 63? They're both divisible by 7! This is where knowing your multiplication tables comes in handy.

• $84 \div 7 = 12$
• $63 \div 7 = 9$

We're down to $12 / 9$. Getting there! Now, 12 and 9 have a common factor of 3:

• $12 \div 3 = 4$
• $9 \div 3 = 3$

Finally, we have $4 / 3$. Can we simplify this any further? Nope! 4 and 3 have no common factors other than 1. So, $4 / 3$ is our fraction in its simplest form.

Alternatively, we could have found the GCF of 336 and 252 right away. The GCF of 336 and 252 is 84. If we divide both the numerator and denominator by 84, we get:

• $336 \div 84 = 4$
• $252 \div 84 = 3$

Which gives us $4 / 3$ directly. This method is faster if you can spot the GCF, but it's perfectly fine to simplify in smaller steps, like we did initially. The important thing is to keep simplifying until you can't anymore.

The Final Answer

Alright, after all that simplifying, we've arrived at our final answer. The simplified form of $6 / 7 \div 36 / 56$ is $4 / 3$. Woohoo! We took a fraction division problem, turned it into a multiplication problem, and then simplified the resulting fraction to its most basic form. That’s a lot of math in one go!

So, looking at our options:

• A) $6 / 8$
• B) $8 / 6$
• C) $2 / 8$
• D) $4 / 3$

The correct answer is D) $4 / 3$.

Key Takeaways

Before we wrap up, let's quickly recap the key steps we took to solve this problem:

1. Dividing by a fraction is the same as multiplying by its reciprocal. This is the golden rule of fraction division.
2. Multiply the numerators and the denominators. Once you've flipped the fraction, it's just a straightforward multiplication problem.
3. Simplify the fraction to its simplest form. This involves finding the greatest common factor (GCF) and dividing both the numerator and denominator by it. You can do this in steps or find the GCF directly.

Understanding these steps will help you tackle any fraction division problem with confidence. Remember, practice makes perfect! The more you work with fractions, the more comfortable you'll become with these concepts.

Practice Problems

Want to test your skills? Try these practice problems:

1. $3 / 4 \div 9 / 16$
2. $1 / 2 \div 5 / 8$
3. $10 / 12 \div 25 / 36$

Work through these problems using the steps we discussed, and you'll be a fraction-dividing pro in no time! Remember, math is like building a house – you need a strong foundation to build on. Mastering fractions is a crucial part of that foundation.

Conclusion

So, there you have it! We've successfully solved the problem $6 / 7 \div 36 / 56$ and arrived at the answer $4 / 3$. We've also walked through the steps involved in dividing fractions and simplifying them. Remember, math isn't about memorizing formulas; it's about understanding the concepts and applying them. By understanding the “why” behind the steps, you'll be able to tackle any math problem that comes your way.

Keep practicing, keep asking questions, and most importantly, keep having fun with math! You've got this!