Simplify The Expression:${ 7 \frac{1}{2} - \left( 2 \frac{1}{2} + 3 \right) \div \frac{3s}{2} }$

Introduction

Mathematical expressions can be complex and daunting, especially when they involve fractions, decimals, and variables. In this article, we will simplify the expression: ${ 7 \frac{1}{2} - \left( 2 \frac{1}{2} + 3 \right) \div \frac{3s}{2} }$. We will break down the expression into smaller parts, evaluate each part, and then combine the results to simplify the expression.

Understanding the Expression

The given expression is a combination of fractions, decimals, and variables. To simplify it, we need to understand the order of operations, which is a set of rules that dictate the order in which mathematical operations should be performed.

The order of operations is as follows:

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.

Breaking Down the Expression

Let's break down the expression into smaller parts and evaluate each part separately.

Part 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is: $2 \frac{1}{2} + 3$.

To evaluate this expression, we need to convert the mixed number $2 \frac{1}{2}$ to an improper fraction. $2 \frac{1}{2}$ is equal to $\frac{5}{2}$.

Now, we can rewrite the expression as: $\frac{5}{2} + 3$.

To add a fraction and a whole number, we need to convert the whole number to a fraction with the same denominator. In this case, we can convert 3 to $\frac{6}{2}$.

Now, we can rewrite the expression as: $\frac{5}{2} + \frac{6}{2}$.

To add fractions with the same denominator, we can simply add the numerators. $\frac{5}{2} + \frac{6}{2} = \frac{11}{2}$.

So, the expression inside the parentheses is equal to $\frac{11}{2}$.

Part 2: Evaluate the Division Operation

The expression is: $\frac{11}{2} \div \frac{3s}{2}$.

To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{3s}{2}$ is $\frac{2}{3s}$.

Now, we can rewrite the expression as: $\frac{11}{2} \times \frac{2}{3s}$.

To multiply fractions, we can multiply the numerators and multiply the denominators. $\frac{11}{2} \times \frac{2}{3s} = \frac{11 \times 2}{2 \times 3s} = \frac{22}{6s}$.

So, the division operation is equal to $\frac{22}{6s}$.

Part 3: Evaluate the Subtraction Operation

The expression is: $7 \frac{1}{2} - \frac{22}{6s}$.

To evaluate this expression, we need to convert the mixed number $7 \frac{1}{2}$ to an improper fraction. $7 \frac{1}{2}$ is equal to $\frac{15}{2}$.

Now, we can rewrite the expression as: $\frac{15}{2} - \frac{22}{6s}$.

To subtract fractions with different denominators, we need to find a common denominator. In this case, the least common multiple of 2 and 6s is 6s.

Now, we can rewrite the expression as: $\frac{15 \times 3s}{2 \times 3s} - \frac{22}{6s}$.

To subtract fractions with the same denominator, we can simply subtract the numerators. $\frac{15 \times 3s}{2 \times 3s} - \frac{22}{6s} = \frac{45s - 22}{6s}$.

So, the final simplified expression is: $\frac{45s - 22}{6s}$.

Conclusion

In this article, we simplified the expression: ${ 7 \frac{1}{2} - \left( 2 \frac{1}{2} + 3 \right) \div \frac{3s}{2} }$. We broke down the expression into smaller parts, evaluated each part separately, and then combined the results to simplify the expression.

The final simplified expression is: $\frac{45s - 22}{6s}$.

This expression can be further simplified by factoring out the common factor of 1 from the numerator and denominator. However, this is not necessary, and the expression is already in its simplest form.

We hope this article has provided a clear and concise explanation of how to simplify complex mathematical expressions. By following the order of operations and breaking down the expression into smaller parts, we can simplify even the most daunting expressions.

Introduction

In our previous article, we simplified the expression: ${ 7 \frac{1}{2} - \left( 2 \frac{1}{2} + 3 \right) \div \frac{3s}{2} }$. We broke down the expression into smaller parts, evaluated each part separately, and then combined the results to simplify the expression.

In this article, we will answer some of the most frequently asked questions about simplifying complex mathematical expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is as follows:

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 do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, you need to follow the order of operations. First, evaluate any exponential expressions, then any multiplication and division operations, and finally any addition and subtraction operations.

Q: How do I evaluate fractions with different denominators?

A: To evaluate fractions with different denominators, you need to find a common denominator. The least common multiple (LCM) of the two denominators is the smallest number that both denominators can divide into evenly.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, you need to find a common denominator. Then, subtract the numerators and keep the common denominator.

Q: How do I simplify complex fractions?

A: To simplify complex fractions, you need to multiply the numerator and denominator by the reciprocal of the denominator. This will eliminate the fraction in the denominator.

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. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

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

A: To convert a mixed number to an improper fraction, you need to multiply the whole number by the denominator and add the numerator. Then, write the result as a fraction with the denominator.

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

A: To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator and write the result as a whole number and a remainder. The remainder becomes the new numerator.

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying complex mathematical expressions. We hope this article has provided a clear and concise explanation of how to simplify complex fractions and expressions.

By following the order of operations and breaking down the expression into smaller parts, we can simplify even the most daunting expressions. We hope this article has been helpful in your understanding of simplifying complex mathematical expressions.

Additional Resources

If you have any further questions or need additional help, please refer to the following resources:

• Khan Academy: Simplifying Complex Fractions
• Mathway: Simplifying Complex Fractions
• Wolfram Alpha: Simplifying Complex Fractions

We hope this article has been helpful in your understanding of simplifying complex mathematical expressions.