Simplify The Expression:$\[ 5 \frac{5}{6} \div \frac{25}{10} \div 3 \frac{2}{5} \\]

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Introduction

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When it comes to simplifying complex mathematical expressions, it's essential to have a solid understanding of the rules and procedures involved. In this article, we'll focus on simplifying the expression $5 \frac{5}{6} \div \frac{25}{10} \div 3 \frac{2}{5}$. We'll break down the problem into manageable steps, using a combination of division, multiplication, and simplification techniques to arrive at the final answer.

Understanding Mixed Numbers

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Before we dive into the problem, let's take a moment to review mixed numbers. A mixed number is a combination of a whole number and a fraction. For example, $5 \frac{5}{6}$ is a mixed number that consists of a whole number (5) and a fraction ($\frac{5}{6}$). When working with mixed numbers, it's essential to remember that the whole number part represents the number of whole units, while the fraction part represents the remaining amount.

Simplifying the Expression

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Now that we have a solid understanding of mixed numbers, let's turn our attention to the expression $5 \frac{5}{6} \div \frac{25}{10} \div 3 \frac{2}{5}$. To simplify this expression, we'll follow the order of operations (PEMDAS):

1. Division: When dividing mixed numbers, we can convert them to improper fractions. To do this, we multiply the whole number part by the denominator and then add the numerator. For example, $5 \frac{5}{6}$ can be converted to an improper fraction as follows:

$$5 \frac{5}{6} = \frac{(5 \times 6) + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6}$$

Similarly, $3 \frac{2}{5}$ can be converted to an improper fraction as follows:

$$3 \frac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$$

Now that we have converted the mixed numbers to improper fractions, we can rewrite the expression as follows:

$$\frac{35}{6} \div \frac{25}{10} \div \frac{17}{5}$$

2. Division: When dividing fractions, we can invert the second fraction and multiply instead. To do this, we multiply the first fraction by the reciprocal of the second fraction. For example:

$$\frac{35}{6} \div \frac{25}{10} = \frac{35}{6} \times \frac{10}{25}$$

Similarly, we can rewrite the second division as follows:

$$\frac{35}{6} \times \frac{10}{25} \div \frac{17}{5} = \frac{35}{6} \times \frac{10}{25} \times \frac{5}{17}$$

3. Multiplication: Now that we have rewritten the expression as a series of multiplications, we can simplify the fractions by multiplying the numerators and denominators separately. For example:

$$\frac{35}{6} \times \frac{10}{25} \times \frac{5}{17} = \frac{(35 \times 10 \times 5)}{(6 \times 25 \times 17)}$$

Simplifying the Numerator and Denominator

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Now that we have multiplied the fractions, we can simplify the numerator and denominator separately. To do this, we can look for common factors between the numbers and cancel them out. For example:

$$\frac{(35 \times 10 \times 5)}{(6 \times 25 \times 17)} = \frac{1750}{2550}$$

We can simplify the numerator and denominator by dividing both numbers by their greatest common factor (GCF), which is 50. For example:

$$\frac{1750}{2550} = \frac{(1750 \div 50)}{(2550 \div 50)} = \frac{35}{51}$$

Conclusion

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In this article, we've simplified the expression $5 \frac{5}{6} \div \frac{25}{10} \div 3 \frac{2}{5}$ using a combination of division, multiplication, and simplification techniques. We've converted the mixed numbers to improper fractions, inverted the second fraction and multiplied instead, and simplified the fractions by multiplying the numerators and denominators separately. Finally, we've simplified the numerator and denominator by dividing both numbers by their greatest common factor (GCF). The final answer is $\frac{35}{51}$.

Frequently Asked Questions

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Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

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

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and then add the numerator. For example:

$$5 \frac{5}{6} = \frac{(5 \times 6) + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6}$$

Q: How do I simplify a fraction?

A: To simplify a fraction, look for common factors between the numerator and denominator and cancel them out. For example:

$$\frac{35}{51} = \frac{(35 \div 1)}{(51 \div 1)} = \frac{35}{51}$$

Note that in this case, there are no common factors between the numerator and denominator, so the fraction cannot be simplified further.

Q: What is the greatest common factor (GCF)?

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

$$\frac{35}{51} = \frac{(35 \div 1)}{(51 \div 1)} = \frac{35}{51}$$

In this case, the GCF is 1, since there are no common factors between the numerator and denominator.

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Frequently Asked Questions

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Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

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

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and then add the numerator. For example:

$$5 \frac{5}{6} = \frac{(5 \times 6) + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6}$$

Q: How do I simplify a fraction?

A: To simplify a fraction, look for common factors between the numerator and denominator and cancel them out. For example:

$$\frac{35}{51} = \frac{(35 \div 1)}{(51 \div 1)} = \frac{35}{51}$$

Note that in this case, there are no common factors between the numerator and denominator, so the fraction cannot be simplified further.

Q: What is the greatest common factor (GCF)?

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

$$\frac{35}{51} = \frac{(35 \div 1)}{(51 \div 1)} = \frac{35}{51}$$

In this case, the GCF is 1, since there are no common factors between the numerator and denominator.

Q: How do I divide mixed numbers?

A: To divide mixed numbers, convert them to improper fractions and then invert the second fraction and multiply instead. For example:

$$5 \frac{5}{6} \div \frac{25}{10} = \frac{35}{6} \times \frac{10}{25}$$

Q: How do I multiply mixed numbers?

A: To multiply mixed numbers, convert them to improper fractions and then multiply the numerators and denominators separately. For example:

$$5 \frac{5}{6} \times 3 \frac{2}{5} = \frac{35}{6} \times \frac{17}{5}$$

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction with a numerator that is greater than or equal to the denominator.

Q: How do I add and subtract mixed numbers?

A: To add and subtract mixed numbers, convert them to improper fractions and then add or subtract the numerators and denominators separately. For example:

$$5 \frac{5}{6} + 3 \frac{2}{5} = \frac{35}{6} + \frac{17}{5}$$

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is important because it helps to:

• Reduce the complexity of the expression
• Make it easier to understand and work with
• Avoid errors and mistakes
• Improve the accuracy of calculations

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, follow these steps:

1. Re-read the problem: Make sure you understand what the problem is asking for.
2. Check your calculations: Verify that your calculations are correct and that you have not made any mistakes.
3. Simplify the expression: Simplify the expression using the rules and procedures you have learned.
4. Check your answer: Check your answer to make sure it is correct and that you have not made any mistakes.

By following these steps, you can ensure that your work is accurate and that you have not made any mistakes.

Conclusion

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In this article, we've simplified the expression $5 \frac{5}{6} \div \frac{25}{10} \div 3 \frac{2}{5}$ using a combination of division, multiplication, and simplification techniques. We've also answered some frequently asked questions about simplifying expressions, including how to convert mixed numbers to improper fractions, how to simplify fractions, and how to check your work when simplifying expressions. By following the steps and procedures outlined in this article, you can simplify complex expressions and improve your understanding of mathematical concepts.