Multiply And Divide Rational Numbers Worksheet

Multiplying and Dividing Rational Numbers: A Comprehensive Worksheet Guide

Understanding how to multiply and divide rational numbers is a fundamental skill in mathematics, crucial for success in algebra and beyond. This comprehensive guide will walk you through the concepts, provide step-by-step examples, and offer a wealth of practice problems to solidify your understanding. We'll cover everything from the basics of rational numbers to tackling more complex problems, ensuring you gain a strong grasp of this essential mathematical skill.

What are Rational Numbers?

Before diving into multiplication and division, let's establish a clear understanding of what rational numbers are. A rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. This includes:

• Integers: Whole numbers (positive, negative, and zero). Examples: -3, 0, 5. These can be expressed as fractions (e.g., 5/1).
• Fractions: Numbers expressed as a ratio of two integers. Examples: 1/2, -3/4, 7/10.
• Terminating Decimals: Decimals that end after a finite number of digits. Examples: 0.25 (which is 1/4), 0.75 (which is 3/4), -0.5 (which is -1/2).
• Repeating Decimals: Decimals that have a pattern of digits that repeats infinitely. Examples: 0.333... (which is 1/3), 0.666... (which is 2/3), 0.142857142857... (which is 1/7).

Numbers that are NOT rational numbers are called irrational numbers. These include numbers like π (pi) and √2 (the square root of 2), which cannot be expressed as a fraction of two integers.

Multiplying Rational Numbers

Multiplying rational numbers involves multiplying the numerators (top numbers) together and multiplying the denominators (bottom numbers) together. Here's the process:

(a/b) * (c/d) = (a * c) / (b * d)

Step-by-Step Example 1:

Multiply (2/3) * (4/5)

1. Multiply the numerators: 2 * 4 = 8
2. Multiply the denominators: 3 * 5 = 15
3. Simplify the fraction (if possible): The resulting fraction is 8/15, which cannot be simplified further.

Therefore, (2/3) * (4/5) = 8/15

Step-by-Step Example 2 (Involving Negative Numbers):

Multiply (-3/4) * (2/7)

1. Multiply the numerators: -3 * 2 = -6
2. Multiply the denominators: 4 * 7 = 28
3. Simplify the fraction: -6/28 can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 2. This gives us -3/14.

Therefore, (-3/4) * (2/7) = -3/14

Step-by-Step Example 3 (Mixed Numbers):

Multiply 2 1/2 * 3 1/3

1. Convert mixed numbers to improper fractions: 2 1/2 = 5/2 and 3 1/3 = 10/3
2. Multiply the improper fractions: (5/2) * (10/3) = (5 * 10) / (2 * 3) = 50/6
3. Simplify the fraction: 50/6 can be simplified to 25/3
4. Convert back to a mixed number (optional): 25/3 = 8 1/3

Therefore, 2 1/2 * 3 1/3 = 8 1/3

Dividing Rational Numbers

Dividing rational numbers is similar to multiplying, but involves an extra step: you need to invert (reciprocate) the second fraction (the divisor) and then multiply.

(a/b) ÷ (c/d) = (a/b) * (d/c)

Step-by-Step Example 1:

Divide (3/4) ÷ (2/5)

1. Invert the second fraction (the divisor): The reciprocal of 2/5 is 5/2.
2. Multiply the fractions: (3/4) * (5/2) = (3 * 5) / (4 * 2) = 15/8

Therefore, (3/4) ÷ (2/5) = 15/8

Step-by-Step Example 2 (Involving Negative Numbers):

Divide (-5/6) ÷ (1/3)

1. Invert the second fraction: The reciprocal of 1/3 is 3/1 (or simply 3).
2. Multiply the fractions: (-5/6) * (3/1) = (-5 * 3) / (6 * 1) = -15/6
3. Simplify the fraction: -15/6 can be simplified to -5/2

Therefore, (-5/6) ÷ (1/3) = -5/2

Step-by-Step Example 3 (Mixed Numbers):

Divide 4 1/2 ÷ 1 1/4

1. Convert mixed numbers to improper fractions: 4 1/2 = 9/2 and 1 1/4 = 5/4
2. Invert the second fraction: The reciprocal of 5/4 is 4/5.
3. Multiply the fractions: (9/2) * (4/5) = (9 * 4) / (2 * 5) = 36/10
4. Simplify the fraction: 36/10 simplifies to 18/5
5. Convert to a mixed number: 18/5 = 3 3/5

Therefore, 4 1/2 ÷ 1 1/4 = 3 3/5

Practice Problems: Multiplying and Dividing Rational Numbers

Now it's time to put your knowledge into practice. Try solving these problems. Remember to show your work step-by-step.

Multiplication:

1. (1/2) * (3/4)
2. (-2/3) * (5/6)
3. (4/7) * (-7/8)
4. 1 1/3 * 2 1/2
5. (-3 1/4) * (-2/5)

Division:

1. (2/5) ÷ (1/3)
2. (-3/4) ÷ (2/5)
3. (7/8) ÷ (-1/2)
4. 2 1/2 ÷ 1 1/4
5. (-3 1/3) ÷ (-2/3)

Advanced Concepts and Applications

Once you've mastered the basics, you can explore more advanced concepts and applications of multiplying and dividing rational numbers:

• Order of Operations (PEMDAS/BODMAS): Remember to follow the order of operations when solving problems that involve multiple operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right).
• Complex Fractions: Fractions containing fractions in their numerators and/or denominators. Simplify these by working with the numerator and denominator separately.
• Solving Equations: You'll use multiplication and division of rational numbers to solve algebraic equations.
• Real-world applications: These mathematical skills are used extensively in various fields like physics, engineering, finance, and cooking! Think of ratios, proportions, and scaling.

Conclusion

Mastering the multiplication and division of rational numbers is a crucial step in your mathematical journey. Through consistent practice and a solid understanding of the concepts, you'll build a strong foundation for more advanced mathematical topics. Remember to break down problems into manageable steps, and don't hesitate to review the examples and explanations provided above. With dedicated effort, you'll confidently tackle any rational number problem!