Which Of The Following Is Equivalent To The Expression:$ -2\left(36 - 60 \div \frac{1}{2}\right) $

Understanding the Expression

The given expression involves a combination of arithmetic operations, including subtraction, division, and multiplication. To simplify this expression, we need to follow the order of operations (PEMDAS), which dictates that we perform division before subtraction.

Breaking Down the Expression

Let's break down the expression step by step:

1. Division: The expression contains the term $60 \div \frac{1}{2}$. To evaluate this, we need to invert the divisor and multiply: $60 \div \frac{1}{2} = 60 \times 2 = 120$.
2. Subtraction: Now that we have the result of the division, we can rewrite the expression as $-2(36 - 120)$.
3. Subtraction inside the parentheses: Evaluating the expression inside the parentheses, we get $36 - 120 = -84$.
4. Multiplication: Finally, we multiply the result by $-2$: $-2(-84) = 168$.

Equivalent Expressions

Now that we have simplified the original expression, let's examine the equivalent expressions that could be written in a different form.

Expression 1: $-2\left(36 - 60 \div \frac{1}{2}\right)$

This is the original expression, which we have already simplified to $168$.

Expression 2: $-2\left(36 - 60 \times 2\right)$

This expression is equivalent to the original expression, as the division and multiplication operations are commutative.

Expression 3: $-2\left(36 - \frac{60}{\frac{1}{2}}\right)$

This expression is also equivalent to the original expression, as the division by a fraction is equivalent to multiplication by its reciprocal.

Expression 4: $-2\left(36 - 120\right)$

This expression is equivalent to the original expression, as the subtraction inside the parentheses is evaluated first.

Conclusion

In conclusion, the equivalent expressions to the given expression $-2\left(36 - 60 \div \frac{1}{2}\right)$ are:

• $-2\left(36 - 60 \times 2\right)$
• $-2\left(36 - \frac{60}{\frac{1}{2}}\right)$
• $-2\left(36 - 120\right)$

All of these expressions simplify to the same result, $168$.

Frequently Asked Questions

• Q: What is the order of operations (PEMDAS)? A: The order of operations is a set of rules that dictate the order in which arithmetic operations should be performed. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Q: How do I evaluate an expression with multiple operations? A: To evaluate an expression with multiple operations, follow the order of operations (PEMDAS). First, evaluate any expressions inside parentheses. Next, evaluate any exponents. Finally, evaluate any multiplication and division operations from left to right, followed by any addition and subtraction operations from left to right.
• Q: What is the difference between division and multiplication? A: Division and multiplication are both arithmetic operations that involve numbers. However, division involves dividing one number by another, while multiplication involves multiplying one number by another.
  

Understanding Equivalent Expressions

Equivalent expressions are mathematical expressions that have the same value, but may be written in different forms. In the previous article, we explored the equivalent expressions to the given expression $-2\left(36 - 60 \div \frac{1}{2}\right)$. In this article, we will answer some frequently asked questions (FAQs) on equivalent expressions.

Q&A on Equivalent Expressions

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical expression that has the same value as another expression, but may be written in a different form.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can simplify both expressions and compare their values. If the values are the same, then the expressions are equivalent.

Q: What are some common ways to write equivalent expressions?

A: Some common ways to write equivalent expressions include:

• Using different operations (e.g., addition vs. subtraction)
• Using different order of operations (e.g., PEMDAS vs. BODMAS)
• Using different mathematical properties (e.g., commutative, associative, distributive)
• Using different algebraic manipulations (e.g., factoring, expanding)

Q: Can equivalent expressions have different variables?

A: Yes, equivalent expressions can have different variables. For example, the expressions $x + 3$ and $y + 3$ are equivalent, even though they have different variables.

Q: Can equivalent expressions have different constants?

A: Yes, equivalent expressions can have different constants. For example, the expressions $2x + 5$ and $3x + 2$ are equivalent, even though they have different constants.

Q: How do I write an equivalent expression for a given expression?

A: To write an equivalent expression for a given expression, you can use the following steps:

1. Simplify the given expression by combining like terms and applying mathematical properties.
2. Use algebraic manipulations (e.g., factoring, expanding) to rewrite the expression in a different form.
3. Use different operations (e.g., addition vs. subtraction) to rewrite the expression in a different form.
4. Use different order of operations (e.g., PEMDAS vs. BODMAS) to rewrite the expression in a different form.

Q: What are some common mistakes to avoid when writing equivalent expressions?

A: Some common mistakes to avoid when writing equivalent expressions include:

• Not simplifying the expression before rewriting it
• Not using the correct order of operations
• Not applying mathematical properties correctly
• Not using algebraic manipulations correctly

Conclusion

In conclusion, equivalent expressions are mathematical expressions that have the same value, but may be written in different forms. By understanding how to write equivalent expressions and avoiding common mistakes, you can simplify complex expressions and solve mathematical problems more efficiently.

Additional Resources

• For more information on equivalent expressions, see the following resources:

• Khan Academy: Equivalent Expressions
• Mathway: Equivalent Expressions
• Wolfram Alpha: Equivalent Expressions

Frequently Asked Questions (FAQs) on Equivalent Expressions

• Q: What is the difference between equivalent expressions and equivalent equations? A: Equivalent expressions are mathematical expressions that have the same value, while equivalent equations are mathematical equations that have the same solution.
• Q: Can equivalent expressions have different units? A: Yes, equivalent expressions can have different units. For example, the expressions $2x + 5$ and $3x + 2$ are equivalent, even though they have different units (e.g., meters vs. feet).
• Q: How do I determine if two expressions are equivalent in a specific context? A: To determine if two expressions are equivalent in a specific context, you need to consider the context and the specific requirements of the problem. For example, in a physics problem, you may need to consider the units of measurement and the physical laws that apply.