1. Select All The Expressions That Are Equivalent To $-15 \div 23$.A. $\frac{15}{-23}$B. $-\frac{15}{23}$C. $\frac{-23}{15}$D. $\frac{15}{23}$E. $\frac{23}{-15}$

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When dealing with division operations, it's essential to understand the concept of equivalent expressions. Equivalent expressions are mathematical expressions that represent the same value or result. In the context of division, equivalent expressions can be obtained by applying the rules of arithmetic operations, such as the order of operations (PEMDAS) and the properties of division.

The Given Expression: −15÷23-15 \div 23

The given expression is −15÷23-15 \div 23. To find equivalent expressions, we need to apply the rules of arithmetic operations and the properties of division.

Applying the Rules of Arithmetic Operations

When dividing two numbers, we can apply the following rules:

  • The order of operations (PEMDAS) states that division should be performed before any other operations.
  • The property of division states that dividing a negative number by a positive number results in a negative quotient.

Equivalent Expressions

Using the rules of arithmetic operations, we can rewrite the given expression as:

−15÷23=−1523-15 \div 23 = \frac{-15}{23}

This is because dividing a negative number by a positive number results in a negative quotient.

Option Analysis

Let's analyze the given options:

A. 15−23\frac{15}{-23}: This option is incorrect because it represents the reciprocal of the given expression, not an equivalent expression.

B. −1523-\frac{15}{23}: This option is correct because it represents the same value as the given expression.

C. −2315\frac{-23}{15}: This option is incorrect because it represents the reciprocal of the given expression, not an equivalent expression.

D. 1523\frac{15}{23}: This option is incorrect because it represents the reciprocal of the given expression, not an equivalent expression.

E. 23−15\frac{23}{-15}: This option is incorrect because it represents the reciprocal of the given expression, not an equivalent expression.

Conclusion

In conclusion, the equivalent expressions to −15÷23-15 \div 23 are:

  • −1523\frac{-15}{23}
  • −1523-\frac{15}{23}

These expressions represent the same value as the given expression and can be obtained by applying the rules of arithmetic operations and the properties of division.

Final Answer

The correct options are B. −1523-\frac{15}{23}.

Additional Examples

To further illustrate the concept of equivalent expressions in division, let's consider a few more examples:

  • −24÷16=−2416=−2416-24 \div 16 = \frac{-24}{16} = -\frac{24}{16}
  • −36÷18=−3618=−3618-36 \div 18 = \frac{-36}{18} = -\frac{36}{18}

In each of these examples, we can see that the equivalent expressions are obtained by applying the rules of arithmetic operations and the properties of division.

Common Mistakes

When dealing with division operations, it's essential to avoid common mistakes, such as:

  • Confusing the order of operations (PEMDAS)
  • Failing to apply the properties of division
  • Not considering the signs of the numbers involved

By being aware of these common mistakes, we can ensure that we obtain the correct equivalent expressions in division.

Real-World Applications

The concept of equivalent expressions in division has numerous real-world applications, such as:

  • Calculating percentages and ratios
  • Solving problems involving rates and proportions
  • Analyzing data and making informed decisions

By understanding equivalent expressions in division, we can develop a deeper understanding of mathematical concepts and apply them to real-world problems.

Conclusion

Q: What is an equivalent expression in division?

A: An equivalent expression in division is a mathematical expression that represents the same value or result as the original expression. Equivalent expressions can be obtained by applying the rules of arithmetic operations and the properties of division.

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

A: To determine if two expressions are equivalent, you can apply the rules of arithmetic operations and the properties of division. You can also use the following steps:

  1. Simplify both expressions by applying the order of operations (PEMDAS).
  2. Check if the expressions have the same value or result.
  3. If the expressions have the same value or result, they are equivalent.

Q: What are some common mistakes to avoid when working with equivalent expressions in division?

A: Some common mistakes to avoid when working with equivalent expressions in division include:

  • Confusing the order of operations (PEMDAS)
  • Failing to apply the properties of division
  • Not considering the signs of the numbers involved
  • Not simplifying expressions before comparing them

Q: How do I apply the rules of arithmetic operations to find equivalent expressions in division?

A: To apply the rules of arithmetic operations to find equivalent expressions in division, you can follow these steps:

  1. Simplify the expression by applying the order of operations (PEMDAS).
  2. Apply the properties of division, such as dividing a negative number by a positive number.
  3. Simplify the expression further by combining like terms.

Q: What are some real-world applications of equivalent expressions in division?

A: Equivalent expressions in division have numerous real-world applications, such as:

  • Calculating percentages and ratios
  • Solving problems involving rates and proportions
  • Analyzing data and making informed decisions
  • Developing mathematical models to solve real-world problems

Q: How do I use equivalent expressions in division to solve problems?

A: To use equivalent expressions in division to solve problems, you can follow these steps:

  1. Read and understand the problem.
  2. Identify the key elements of the problem, such as the numbers and operations involved.
  3. Apply the rules of arithmetic operations and the properties of division to find equivalent expressions.
  4. Simplify the expressions and solve the problem.

Q: What are some examples of equivalent expressions in division?

A: Some examples of equivalent expressions in division include:

  • −15÷23=−1523=−1523-15 \div 23 = \frac{-15}{23} = -\frac{15}{23}
  • −24÷16=−2416=−2416-24 \div 16 = \frac{-24}{16} = -\frac{24}{16}
  • −36÷18=−3618=−3618-36 \div 18 = \frac{-36}{18} = -\frac{36}{18}

Q: How do I check if an expression is equivalent to another expression?

A: To check if an expression is equivalent to another expression, you can follow these steps:

  1. Simplify both expressions by applying the order of operations (PEMDAS).
  2. Check if the expressions have the same value or result.
  3. If the expressions have the same value or result, they are equivalent.

Q: What are some tips for working with equivalent expressions in division?

A: Some tips for working with equivalent expressions in division include:

  • Always simplify expressions before comparing them.
  • Apply the rules of arithmetic operations and the properties of division carefully.
  • Check your work to ensure that the expressions are equivalent.
  • Practice working with equivalent expressions in division to develop your skills.