Alright, guys, let's break down this math problem step by step so you can easily understand how to solve it. We're tackling the calculation: 6/5 ÷ 1/6 × 7/5. It looks a bit intimidating at first, but don't worry, we'll make it super clear.

Understanding the Order of Operations

Before we dive in, remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our case, we have division and multiplication. Since they're on the same level of priority, we perform them from left to right. This is crucial for getting the correct answer. Many people get tripped up by not following this order, so keep it in mind!

First, we'll handle the division: 6/5 ÷ 1/6. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/6 is 6/1 (or just 6). So, we rewrite the division as a multiplication:

6/5 ÷ 1/6 = 6/5 × 6/1

Now, multiply the numerators (top numbers) and the denominators (bottom numbers):

(6 × 6) / (5 × 1) = 36/5

So, the result of the division is 36/5. Great, we're halfway there!

Next, we take this result and multiply it by the remaining fraction, 7/5:

36/5 × 7/5

Again, multiply the numerators and the denominators:

(36 × 7) / (5 × 5) = 252/25

Therefore, the final answer is 252/25. You can leave it as an improper fraction, or convert it to a mixed number if you prefer. To convert it to a mixed number, divide 252 by 25. The quotient is 10, and the remainder is 2. So, the mixed number is 10 2/25.

Why Understanding Fractions Matters

Fractions are fundamental in math and appear everywhere, from baking recipes to calculating proportions in science. Knowing how to confidently handle fractions, especially in combination with other operations, is a super valuable skill. It builds a solid foundation for more advanced mathematical concepts. So, practice makes perfect! The more you work with fractions, the more comfortable you'll become.

Common Mistakes to Avoid

• Forgetting the Order of Operations: Always remember PEMDAS (or BODMAS, depending on where you learned math). Doing operations in the wrong order will lead to incorrect results.
• Incorrectly Finding the Reciprocal: Make sure you flip the fraction correctly when dividing. The reciprocal of a/b is b/a.
• Not Simplifying: Although 252/25 is a correct answer, always check if you can simplify the fraction further. In this case, it can’t be simplified, but always be on the lookout!
• Calculator Dependency: While calculators are helpful, try to understand the underlying concepts. This way, you can catch errors and have a better grasp of what you're doing.

Practicing More Problems

To really nail this down, try practicing similar problems. For example:

• 8/3 ÷ 2/5 × 1/2
• 10/7 × 3/4 ÷ 5/6
• 12/5 ÷ 4/3 × 2/7

Work through these problems step by step, and check your answers. If you get stuck, review the steps we covered earlier. Remember, each problem you solve builds your confidence and skills.

Real-World Applications

You might be wondering, "Where would I ever use this in real life?" Well, calculating proportions is a common task in many fields. Here are a few examples:

• Cooking: Adjusting recipe quantities. If a recipe calls for 2/3 cup of flour and you want to make half the recipe, you'll need to calculate half of 2/3.
• Construction: Measuring materials. If you're building a fence and each section requires 5/8 of a meter of wood, you'll need to calculate how much wood you need for multiple sections.
• Finance: Calculating investment returns. Understanding fractions and percentages is crucial for understanding how your investments are performing.

Conclusion

So, to recap, we successfully solved the calculation 6/5 ÷ 1/6 × 7/5 by following the order of operations and understanding how to divide and multiply fractions. The final answer is 252/25 (or 10 2/25 as a mixed number). Remember to practice regularly, avoid common mistakes, and understand the real-world applications. You've got this!

Now that we've thoroughly covered this particular problem, let's zoom out and discuss some broader strategies for mastering fraction arithmetic. This isn't just about memorizing steps; it's about building a strong conceptual understanding. Mastering fraction arithmetic involves more than just memorizing rules; it requires a solid conceptual understanding and consistent practice. Here’s a more in-depth look at how to achieve this:

Building a Conceptual Foundation

Understanding what fractions represent is the first key step. Think of a fraction as a part of a whole. For example, 3/4 represents three parts out of four equal parts. This simple visualization can make operations like addition and subtraction more intuitive. Using visual aids such as pie charts or bar models can greatly enhance this understanding, especially for beginners. Manipulatives, like fraction bars or circles, allow you to physically represent fractions and perform operations. This tactile learning can be particularly helpful for grasping the concept of equivalent fractions and simplifying fractions.

Mastering Equivalent Fractions

Equivalent fractions are fractions that represent the same value but have different numerators and denominators (e.g., 1/2 = 2/4 = 3/6). The ability to find equivalent fractions is crucial for adding and subtracting fractions with different denominators. To find equivalent fractions, multiply or divide both the numerator and the denominator by the same non-zero number. For example, to convert 1/3 to an equivalent fraction with a denominator of 6, multiply both the numerator and the denominator by 2, resulting in 2/6.

Adding and Subtracting Fractions

To add or subtract fractions, they must have the same denominator. If they don't, you need to find a common denominator first. The least common denominator (LCD) is the smallest common multiple of the denominators. Once you have a common denominator, simply add or subtract the numerators and keep the denominator the same. For example, to add 1/4 and 2/5, the LCD is 20. Convert 1/4 to 5/20 and 2/5 to 8/20. Then, add the numerators: 5/20 + 8/20 = 13/20.

Multiplying Fractions

Multiplying fractions is straightforward: simply multiply the numerators and multiply the denominators. Simplify the resulting fraction if possible. For example, to multiply 2/3 by 3/4: (2 × 3) / (3 × 4) = 6/12, which simplifies to 1/2. Cross-cancellation can simplify the process, especially when dealing with larger numbers. Look for common factors between the numerator of one fraction and the denominator of the other, and cancel them out before multiplying.

Dividing Fractions

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 2/3 is 3/2. To divide 1/2 by 2/3, multiply 1/2 by 3/2: (1/2) ÷ (2/3) = (1/2) × (3/2) = 3/4. Understanding why this works can be achieved by thinking about division as the inverse of multiplication. When you divide by a fraction, you are essentially asking, "How many of this fraction fit into the number I'm dividing?"

Practicing Regularly

Consistent practice is key to mastering fraction arithmetic. Start with simple problems and gradually increase the difficulty. Work through a variety of problems involving different operations and combinations of operations. Use online resources, textbooks, and worksheets to find practice problems. The more you practice, the more comfortable and confident you'll become with fractions. Keep a record of your work, noting any mistakes you make and the steps you took to correct them. This will help you identify areas where you need more practice and track your progress over time.

Utilizing Technology

There are many online tools and apps available that can help you practice and improve your skills in fraction arithmetic. These tools can provide instant feedback, generate practice problems, and offer step-by-step solutions. Some tools also offer visual representations of fractions, which can be helpful for understanding the concepts. Don't rely solely on calculators to solve problems. While calculators can be helpful for checking your work, it's important to be able to perform the calculations manually to develop a strong understanding of the underlying concepts.

Seeking Help When Needed

Don't hesitate to ask for help if you're struggling with fraction arithmetic. Talk to your teacher, a tutor, or a friend who is good at math. There are also many online resources available, such as video tutorials and forums where you can ask questions and get help from others. Sometimes, a different explanation or perspective can make all the difference. Remember, everyone learns at their own pace. Don't get discouraged if you don't understand something right away. Keep practicing and seeking help when needed, and you'll eventually master fraction arithmetic.