Practice Multiplying Negative Rational NumbersStudy The Example Showing How To Multiply Negative Rational Numbers. Then Solve Problems 1-5.ExampleWhat Is $-6\left(-8 \frac{1}{2}\right$\]?- Convert $-8 \frac{1}{2}$ To An Improper

by ADMIN 229 views

Introduction

Multiplying negative rational numbers can be a daunting task for many students, but with the right approach and practice, it can become a breeze. In this article, we will delve into the world of negative rational numbers and explore the steps involved in multiplying them. We will also provide a step-by-step example and practice problems to help you master this essential math skill.

Understanding Negative Rational Numbers

Before we dive into multiplying negative rational numbers, it's essential to understand what negative rational numbers are. A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A negative rational number is a rational number that is less than zero.

Example: Converting a Mixed Number to an Improper Fraction

To multiply negative rational numbers, we first need to convert the mixed number to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Let's take the example of −812-8 \frac{1}{2}. To convert this mixed number to an improper fraction, we multiply the denominator (2) by the whole number (8) and then add the numerator (1).

−812=−(2×8)+12=−16+12=−172-8 \frac{1}{2} = -\frac{(2 \times 8) + 1}{2} = -\frac{16 + 1}{2} = -\frac{17}{2}

Multiplying Negative Rational Numbers

Now that we have converted the mixed number to an improper fraction, we can multiply the two negative rational numbers.

−6(−812)=−6(−172)-6\left(-8 \frac{1}{2}\right) = -6\left(-\frac{17}{2}\right)

To multiply two negative rational numbers, we multiply the numerators and denominators separately and then apply the rule of signs.

Rule of Signs

When multiplying two negative rational numbers, we apply the rule of signs, which states that two negative numbers multiplied together result in a positive number.

−6(−172)=(−6)×(−17)2=1022=51-6\left(-\frac{17}{2}\right) = \frac{(-6) \times (-17)}{2} = \frac{102}{2} = 51

Practice Problems

Now that we have mastered the art of multiplying negative rational numbers, let's practice with some problems.

Problem 1

What is −3(−423)-3\left(-4 \frac{2}{3}\right)?

Problem 2

What is −2(−514)-2\left(-5 \frac{1}{4}\right)?

Problem 3

What is −4(−634)-4\left(-6 \frac{3}{4}\right)?

Problem 4

What is −5(−712)-5\left(-7 \frac{1}{2}\right)?

Problem 5

What is −6(−823)-6\left(-8 \frac{2}{3}\right)?

Solutions

Problem 1

−3(−423)=−3(−143)=(−3)×(−14)3=423=14-3\left(-4 \frac{2}{3}\right) = -3\left(-\frac{14}{3}\right) = \frac{(-3) \times (-14)}{3} = \frac{42}{3} = 14

Problem 2

−2(−514)=−2(−214)=(−2)×(−21)4=424=10.5-2\left(-5 \frac{1}{4}\right) = -2\left(-\frac{21}{4}\right) = \frac{(-2) \times (-21)}{4} = \frac{42}{4} = 10.5

Problem 3

−4(−634)=−4(−274)=(−4)×(−27)4=1084=27-4\left(-6 \frac{3}{4}\right) = -4\left(-\frac{27}{4}\right) = \frac{(-4) \times (-27)}{4} = \frac{108}{4} = 27

Problem 4

−5(−712)=−5(−152)=(−5)×(−15)2=752=37.5-5\left(-7 \frac{1}{2}\right) = -5\left(-\frac{15}{2}\right) = \frac{(-5) \times (-15)}{2} = \frac{75}{2} = 37.5

Problem 5

−6(−823)=−6(−263)=(−6)×(−26)3=1563=52-6\left(-8 \frac{2}{3}\right) = -6\left(-\frac{26}{3}\right) = \frac{(-6) \times (-26)}{3} = \frac{156}{3} = 52

Conclusion

Q: What is the rule of signs for multiplying negative rational numbers?

A: The rule of signs states that two negative numbers multiplied together result in a positive number. When multiplying two negative rational numbers, we apply this rule to get a positive result.

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

A: To convert a mixed number to an improper fraction, multiply the denominator by the whole number and then add the numerator. For example, to convert −812-8 \frac{1}{2} to an improper fraction, we multiply the denominator (2) by the whole number (8) and then add the numerator (1).

−812=−(2×8)+12=−16+12=−172-8 \frac{1}{2} = -\frac{(2 \times 8) + 1}{2} = -\frac{16 + 1}{2} = -\frac{17}{2}

Q: What is the difference between a rational number and a negative rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A negative rational number is a rational number that is less than zero.

Q: Can I multiply a negative rational number by a positive rational number?

A: Yes, you can multiply a negative rational number by a positive rational number. When multiplying a negative rational number by a positive rational number, the result will be a negative rational number.

Q: How do I multiply two negative rational numbers?

A: To multiply two negative rational numbers, multiply the numerators and denominators separately and then apply the rule of signs. For example, to multiply −6(−812)-6\left(-8 \frac{1}{2}\right), we multiply the numerators and denominators separately and then apply the rule of signs.

−6(−812)=−6(−172)=(−6)×(−17)2=1022=51-6\left(-8 \frac{1}{2}\right) = -6\left(-\frac{17}{2}\right) = \frac{(-6) \times (-17)}{2} = \frac{102}{2} = 51

Q: Can I use a calculator to multiply negative rational numbers?

A: Yes, you can use a calculator to multiply negative rational numbers. However, it's essential to understand the concept and be able to multiply negative rational numbers by hand.

Q: What are some common mistakes to avoid when multiplying negative rational numbers?

A: Some common mistakes to avoid when multiplying negative rational numbers include:

  • Forgetting to apply the rule of signs
  • Not converting mixed numbers to improper fractions
  • Multiplying the numerators and denominators incorrectly
  • Not simplifying the result

Q: How can I practice multiplying negative rational numbers?

A: You can practice multiplying negative rational numbers by working through problems and exercises. You can also use online resources and math apps to practice multiplying negative rational numbers.

Q: What are some real-world applications of multiplying negative rational numbers?

A: Multiplying negative rational numbers has many real-world applications, including:

  • Calculating interest rates and investments
  • Determining the cost of goods and services
  • Understanding the concept of debt and credit
  • Solving problems in physics and engineering

Conclusion

Multiplying negative rational numbers can be a challenging concept, but with practice and the right approach, it can become a breeze. We hope this FAQ article has provided you with a comprehensive understanding of multiplying negative rational numbers and has given you the confidence to tackle more complex math problems.