Critique The Reasoning And Find The Error In The Work Below. Then Show The Correct Calculation.${ \frac{8}{12} \times 6 = 8 \times \frac{1}{12} \times 6 = 8 \times \frac{1}{72} = \frac{8}{72} = \frac{1}{9} }$

Introduction

Mathematics is a precise and logical subject that requires careful attention to detail. However, even the most skilled mathematicians can make errors in their calculations. In this article, we will critique the reasoning and find the error in a given mathematical work. We will then provide the correct calculation.

The Given Work

The given work is as follows:

{ \frac{8}{12} \times 6 = 8 \times \frac{1}{12} \times 6 = 8 \times \frac{1}{72} = \frac{8}{72} = \frac{1}{9} \}

Critique of the Reasoning

At first glance, the given work appears to be a correct calculation. However, upon closer inspection, we can see that there is an error in the reasoning. The error lies in the fact that the given work is trying to simplify the expression $\frac{8}{12} \times 6$ by multiplying the numerator and denominator of the fraction by 6.

However, this is not a valid simplification. When we multiply a fraction by a number, we are essentially multiplying the numerator by that number. In this case, we should be multiplying the numerator (8) by 6, not the denominator (12).

Finding the Error

To find the error, let's go through the calculation step by step.

1. $\frac{8}{12} \times 6$

To simplify this expression, we can multiply the numerator (8) by 6, and the denominator (12) by 1.

$\frac{8}{12} \times 6 = \frac{8 \times 6}{12 \times 1} = \frac{48}{12}$

2. $\frac{48}{12}$

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 12.

$\frac{48}{12} = \frac{48 \div 12}{12 \div 12} = \frac{4}{1} = 4$

Correct Calculation

So, the correct calculation is:

$\frac{8}{12} \times 6 = \frac{48}{12} = 4$

Conclusion

In conclusion, the given work contains an error in the reasoning. The error lies in the fact that the given work is trying to simplify the expression $\frac{8}{12} \times 6$ by multiplying the numerator and denominator of the fraction by 6. However, this is not a valid simplification. The correct calculation is $\frac{8}{12} \times 6 = \frac{48}{12} = 4$.

Common Mistakes in Mathematical Calculations

There are several common mistakes that people make in mathematical calculations. Some of these mistakes include:

• Not following the order of operations: This can lead to incorrect calculations and incorrect answers.
• Not simplifying fractions: This can lead to complex and difficult-to-read calculations.
• Not checking for errors: This can lead to incorrect answers and a lack of confidence in one's calculations.

Tips for Avoiding Common Mistakes

To avoid common mistakes in mathematical calculations, follow these tips:

• Follow the order of operations: This means that you should perform calculations in the correct order, from left to right.
• Simplify fractions: This means that you should simplify fractions as much as possible to make calculations easier.
• Check for errors: This means that you should check your calculations carefully to make sure that they are correct.

Conclusion

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Q: What is the mistake in the given work?

A: The mistake in the given work is that it is trying to simplify the expression $\frac{8}{12} \times 6$ by multiplying the numerator and denominator of the fraction by 6. However, this is not a valid simplification.

Q: Why is the given work incorrect?

A: The given work is incorrect because it is trying to simplify the expression $\frac{8}{12} \times 6$ by multiplying the numerator and denominator of the fraction by 6. However, this is not a valid simplification. When we multiply a fraction by a number, we are essentially multiplying the numerator by that number, not the denominator.

Q: What is the correct calculation for $\frac{8}{12} \times 6$?

A: The correct calculation for $\frac{8}{12} \times 6$ is:

$\frac{8}{12} \times 6 = \frac{48}{12} = 4$

Q: Why is it important to follow the order of operations?

A: It is important to follow the order of operations because it ensures that calculations are performed in the correct order, from left to right. This helps to avoid errors and ensures that calculations are accurate.

Q: What is the order of operations?

A: The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How can I avoid making mistakes in my mathematical calculations?

A: To avoid making mistakes in your mathematical calculations, follow these tips:

• Follow the order of operations.
• Simplify fractions as much as possible.
• Check your calculations carefully to make sure that they are correct.

Q: What are some common mistakes that people make in mathematical calculations?

A: Some common mistakes that people make in mathematical calculations include:

• Not following the order of operations.
• Not simplifying fractions.
• Not checking for errors.

Q: How can I simplify fractions?

A: To simplify fractions, you can divide both the numerator and denominator by their greatest common divisor (GCD). You can also use a calculator to simplify fractions.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction without leaving a remainder.

Q: How can I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use a calculator or a GCD algorithm. You can also use a factor tree to find the GCD.

Q: What is a factor tree?

A: A factor tree is a diagram that shows the factors of a number. It is a useful tool for finding the GCD of two numbers.

Conclusion

In conclusion, the given work contains an error in the reasoning. The error lies in the fact that the given work is trying to simplify the expression $\frac{8}{12} \times 6$ by multiplying the numerator and denominator of the fraction by 6. However, this is not a valid simplification. The correct calculation is $\frac{8}{12} \times 6 = \frac{48}{12} = 4$. By following the tips for avoiding common mistakes, you can avoid making similar errors in your own mathematical calculations.