Hey guys, let's break down this math problem together! We're going to tackle the calculation: 6/5 ÷ 1/6 × 7/5. It might look a bit intimidating at first, but don't worry, we'll go through it step by step. Understanding the order of operations and how to handle fractions is key to solving this accurately. So, grab your calculators or a piece of paper, and let’s get started!

Understanding the Order of Operations

Before we dive into the specifics of this problem, let's quickly recap the order of operations. You might have heard of it as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). The main thing to remember is that multiplication and division are performed from left to right, and the same goes for addition and subtraction. In our case, we have division and multiplication, so we’ll perform them in the order they appear.

First, we have 6/5 ÷ 1/6. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/6 is 6/1, which is simply 6. So, we rewrite the division as multiplication: 6/5 × 6. This gives us (6 × 6) / 5 = 36/5. Now, we have 36/5 × 7/5. To multiply these two fractions, we multiply the numerators (top numbers) and the denominators (bottom numbers). That is, (36 × 7) / (5 × 5) = 252/25. Thus, 6/5 ÷ 1/6 × 7/5 = 252/25. To further simplify the result, let's convert the improper fraction 252/25 into a mixed number. To do this, we divide 252 by 25. The quotient is 10, and the remainder is 2. Therefore, 252/25 = 10 2/25. So, the final result is 10 2/25.

Step-by-Step Calculation

Let's walk through the calculation step-by-step to make sure everything is crystal clear:

1. Divide 6/5 by 1/6:

   • To divide by a fraction, multiply by its reciprocal.
   • 6/5 ÷ 1/6 = 6/5 × 6/1 = 36/5

2. Multiply the result by 7/5:

   • 36/5 × 7/5 = (36 × 7) / (5 × 5) = 252/25

3. Simplify the fraction:

   • 252/25 can be left as an improper fraction, or converted to a mixed number.
   • 252 ÷ 25 = 10 with a remainder of 2.
   • So, 252/25 = 10 2/25

Therefore, 6/5 ÷ 1/6 × 7/5 = 252/25 = 10 2/25. Each step is crucial to arriving at the correct answer, and understanding why we perform each operation in that specific order helps reinforce the underlying mathematical principles. Keep practicing, and these kinds of calculations will become second nature!

Common Mistakes to Avoid

When dealing with calculations like this, it’s easy to make a few common mistakes. Here are some pitfalls to watch out for:

• Forgetting the Order of Operations: Always remember PEMDAS/BODMAS. In this case, division and multiplication are done from left to right.
• Incorrectly Finding the Reciprocal: Make sure you flip the fraction correctly when dividing. For example, the reciprocal of 1/6 is 6/1, not the other way around.
• Multiplying Before Dividing: In our problem, we divide 6/5 by 1/6 first because it comes first from left to right. Doing multiplication before division would give a completely different answer.
• Arithmetic Errors: Double-check your multiplication and division to avoid simple calculation mistakes. It's easy to slip up, especially when working quickly.
• Not Simplifying the Fraction: While 252/25 is a correct answer, it's often better to simplify it to 10 2/25, especially if your teacher or the problem requires it. Always look for ways to simplify your answer.

Avoiding these mistakes can significantly improve your accuracy and confidence when solving similar problems. Always double-check your work and take your time to ensure each step is correct.

Alternative Methods to Solve

While the step-by-step method is straightforward, there are alternative approaches you can use to solve this problem. These methods can sometimes simplify the calculation or provide a different perspective.

Method 1: Converting to Decimals (Caution Required)

One approach is to convert the fractions to decimals before performing the operations. This can make the arithmetic easier for some people, but it's crucial to be precise to avoid rounding errors. For instance:

• 6/5 = 1.2
• 1/6 ≈ 0.1667 (repeating decimal)
• 7/5 = 1.4

So, the problem becomes: 1.2 ÷ 0.1667 × 1.4. Performing the division first: 1.2 ÷ 0.1667 ≈ 7.1985. Then, multiply by 1.4: 7.1985 × 1.4 ≈ 10.0779. Converting 10 2/25 to a decimal gives us 10.08. As you can see, the decimal method can introduce slight inaccuracies due to rounding. Therefore, it's often better to stick with fractions for precise calculations.

Method 2: Combining Multiplication and Division

Another way to think about the problem is to rewrite the expression to group the multiplication and division together. Since dividing by a fraction is the same as multiplying by its reciprocal, we can rewrite the entire expression as a series of multiplications:

6/5 ÷ 1/6 × 7/5 = 6/5 × 6/1 × 7/5

Now, you can multiply all the numerators together and all the denominators together:

(6 × 6 × 7) / (5 × 1 × 5) = 252 / 25

This gives us the same result as before, 252/25, which simplifies to 10 2/25. This method can be more efficient for some people because it avoids intermediate steps and combines all the operations into one calculation.

Real-World Applications

You might be wondering, where would you ever use this kind of calculation in real life? Well, understanding how to work with fractions, division, and multiplication is essential in many fields and everyday situations. Here are a few examples:

• Cooking and Baking: Recipes often involve fractions. If you need to double or halve a recipe, you'll need to multiply or divide fractions correctly.
• Construction and Carpentry: Measuring materials and calculating dimensions frequently involve fractions. Accuracy is crucial to ensure a project is completed correctly.
• Finance and Investing: Calculating returns on investments, interest rates, and other financial metrics often requires working with fractions and percentages.
• Engineering and Science: Many scientific formulas and engineering calculations involve fractions and ratios. Precise calculations are essential for accurate results.
• Everyday Life: Splitting bills with friends, calculating discounts at the store, or figuring out how much gas you need for a trip all involve basic arithmetic with fractions and decimals.

By mastering these fundamental mathematical skills, you'll be better equipped to handle a wide range of practical problems and make informed decisions in various aspects of life. So, keep practicing, and don't underestimate the power of a solid understanding of fractions and operations!

Conclusion

Alright, guys! We've successfully tackled the calculation 6/5 ÷ 1/6 × 7/5. Remember, the key is to follow the order of operations (PEMDAS/BODMAS) and understand how to work with fractions. By dividing 6/5 by 1/6 and then multiplying the result by 7/5, we arrived at the answer 252/25, which simplifies to 10 2/25. We also explored common mistakes to avoid and alternative methods to solve the problem.

Understanding these concepts not only helps with math class but also equips you with valuable skills for real-world applications. Whether you're cooking, building, managing finances, or solving scientific problems, a solid grasp of fractions and arithmetic operations is essential. So, keep practicing, stay curious, and embrace the power of math in your everyday life! You've got this!