Evaluate Each Expression. When The Answer Is Not A Whole Number, Write Your Answer As A Fraction.a. $(-4) \cdot (-6$\] B. $4 \div (-6$\] C. $(-24) \cdot \left(-\frac{1}{6}\right$\] D. $\frac{4}{3} \div (-24$\]

In mathematics, expressions can be evaluated using various operations such as multiplication, division, addition, and subtraction. When dealing with fractions and negative numbers, it's essential to understand the rules of operation to arrive at the correct answer. In this article, we will evaluate four different expressions, paying close attention to the rules of multiplication, division, and the order of operations.

Expression a: $(-4) \cdot (-6)$

To evaluate this expression, we need to multiply two negative numbers. When multiplying two negative numbers, the result is always positive. Therefore, $(-4) \cdot (-6) = 24$.

Expression b: $4 \div (-6)$

In this expression, we are dividing a positive number by a negative number. When dividing a positive number by a negative number, the result is always negative. Therefore, $4 \div (-6) = -\frac{2}{3}$.

Expression c: $(-24) \cdot \left(-\frac{1}{6}\right)$

To evaluate this expression, we need to multiply a negative number by a fraction. When multiplying a negative number by a fraction, the result is always negative. We can multiply the numerator and denominator of the fraction by the negative number to simplify the expression. Therefore, $(-24) \cdot \left(-\frac{1}{6}\right) = 4$.

Expression d: $\frac{4}{3} \div (-24)$

In this expression, we are dividing a fraction by a negative number. When dividing a fraction by a negative number, the result is always negative. We can multiply the numerator and denominator of the fraction by the negative number to simplify the expression. Therefore, $\frac{4}{3} \div (-24) = -\frac{1}{18}$.

Understanding the Rules of Operation

When evaluating expressions with fractions and negative numbers, it's essential to understand the rules of operation. Here are some key rules to keep in mind:

• When multiplying two negative numbers, the result is always positive.
• When dividing a positive number by a negative number, the result is always negative.
• When multiplying a negative number by a fraction, the result is always negative.
• When dividing a fraction by a negative number, the result is always negative.

Conclusion

Evaluating expressions with fractions and negative numbers requires a clear understanding of the rules of operation. By following these rules, we can arrive at the correct answer for each expression. In this article, we evaluated four different expressions, paying close attention to the rules of multiplication, division, and the order of operations. By mastering these rules, we can become proficient in evaluating expressions with fractions and negative numbers.

Common Mistakes to Avoid

When evaluating expressions with fractions and negative numbers, there are several common mistakes to avoid. Here are some tips to help you avoid these mistakes:

• Make sure to follow the order of operations (PEMDAS).
• Be careful when multiplying and dividing fractions.
• Pay attention to the signs of the numbers in the expression.
• Use a calculator or work out the problem step-by-step to ensure accuracy.

Practice Exercises

To practice evaluating expressions with fractions and negative numbers, try the following exercises:

• Evaluate the expression $(-3) \cdot (-9)$.
• Evaluate the expression $6 \div (-2)$.
• Evaluate the expression $(-15) \cdot \left(-\frac{1}{4}\right)$.
• Evaluate the expression $\frac{2}{5} \div (-10)$.

By following the rules of operation and practicing with these exercises, you can become proficient in evaluating expressions with fractions and negative numbers.

Real-World Applications

Evaluating expressions with fractions and negative numbers has many real-world applications. Here are a few examples:

• In finance, you may need to calculate the interest on a loan or investment, which involves evaluating expressions with fractions and negative numbers.
• In science, you may need to calculate the speed or distance of an object, which involves evaluating expressions with fractions and negative numbers.
• In engineering, you may need to calculate the stress or strain on a material, which involves evaluating expressions with fractions and negative numbers.

By mastering the rules of operation and practicing with exercises, you can apply these skills to real-world problems and become a proficient mathematician.

Conclusion

In our previous article, we discussed the rules of operation for evaluating expressions with fractions and negative numbers. In this article, we will answer some common questions related to this topic.

Q: What is the difference between multiplying and dividing fractions?

A: When multiplying fractions, we multiply the numerators and denominators separately. When dividing fractions, we invert the second fraction and multiply.

Q: How do I handle negative numbers when multiplying and dividing fractions?

A: When multiplying a negative number by a fraction, the result is always negative. When dividing a fraction by a negative number, the result is always negative.

Q: What is the order of operations for evaluating expressions with fractions and negative numbers?

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 do I simplify fractions when evaluating expressions with fractions and negative numbers?

A: To simplify a fraction, we divide both the numerator and denominator by their greatest common divisor (GCD).

Q: What is the difference between a positive and negative fraction?

A: A positive fraction is a fraction with a positive numerator and denominator. A negative fraction is a fraction with a negative numerator or denominator.

Q: How do I handle fractions with different signs when multiplying and dividing?

A: When multiplying fractions with different signs, the result is always negative. When dividing fractions with different signs, the result is always negative.

Q: What are some common mistakes to avoid when evaluating expressions with fractions and negative numbers?

A: Some common mistakes to avoid include:

• Not following the order of operations.
• Not simplifying fractions.
• Not handling negative numbers correctly.
• Not using a calculator or working out the problem step-by-step.

Q: How can I practice evaluating expressions with fractions and negative numbers?

A: You can practice evaluating expressions with fractions and negative numbers by working out problems on your own or using online resources such as worksheets and practice tests.

Q: What are some real-world applications of evaluating expressions with fractions and negative numbers?

A: Some real-world applications of evaluating expressions with fractions and negative numbers include:

• Finance: Calculating interest on loans or investments.
• Science: Calculating speed or distance of objects.
• Engineering: Calculating stress or strain on materials.

Conclusion

Evaluating expressions with fractions and negative numbers requires a clear understanding of the rules of operation. By following these rules and practicing with exercises, you can become proficient in evaluating expressions with fractions and negative numbers. Whether you're working in finance, science, or engineering, mastering these skills will help you solve real-world problems and become a confident mathematician.

Additional Resources

For more information on evaluating expressions with fractions and negative numbers, check out the following resources:

• Khan Academy: Fractions and Negative Numbers
• Mathway: Fractions and Negative Numbers
• Wolfram Alpha: Fractions and Negative Numbers

By mastering the rules of operation and practicing with exercises, you can become proficient in evaluating expressions with fractions and negative numbers.