How To Do Distributive Property With Fractions

Mastering Distributive Property with Fractions: A Comprehensive Guide

The distributive property is a fundamental concept in mathematics, allowing us to simplify expressions involving multiplication and addition (or subtraction). While straightforward with whole numbers, it can seem more challenging when dealing with fractions. This comprehensive guide will walk you through the process, providing clear explanations, examples, and tips to help you master distributive property with fractions.

Understanding the Distributive Property

The distributive property states that for any numbers a, b, and c:

a(b + c) = ab + ac

This means you can distribute the number 'a' to both 'b' and 'c', multiplying them separately and then adding the results. The same principle applies when subtracting:

a(b - c) = ab - ac

This seemingly simple rule becomes more nuanced when fractions are involved. Let's explore how to apply it effectively.

Distributive Property with Fractions: Step-by-Step Guide

Applying the distributive property with fractions involves the same core principle, but requires careful attention to fraction arithmetic. Let's break it down step-by-step:

Step 1: Identify the Distributive Expression

First, identify the expression where the distributive property can be applied. This will usually involve a fraction multiplied by a sum or difference within parentheses. For instance:

• (1/2)(3/4 + 1/8)
• (2/3)(5/6 - 1/2)
• (-3/5)(2 + 4/5)

Step 2: Distribute the Fraction

Next, distribute the fraction to each term inside the parentheses. Remember to multiply the fraction by each term individually. This often involves multiplying numerators and denominators.

Example 1:

(1/2)(3/4 + 1/8) = (1/2)(3/4) + (1/2)(1/8)

Example 2:

(2/3)(5/6 - 1/2) = (2/3)(5/6) - (2/3)(1/2)

Example 3:

(-3/5)(2 + 4/5) = (-3/5)(2) + (-3/5)(4/5)

Notice how we've effectively broken down the original expression into two simpler multiplication problems.

Step 3: Perform the Multiplication

Now, perform the multiplication for each term. Remember the rules for multiplying fractions: multiply the numerators together and multiply the denominators together.

Example 1 (continued):

(1/2)(3/4) = 3/8 (1/2)(1/8) = 1/16

So the expression becomes: 3/8 + 1/16

Example 2 (continued):

(2/3)(5/6) = 10/18 (which can be simplified to 5/9) (2/3)(1/2) = 2/6 (which can be simplified to 1/3)

So the expression becomes: 5/9 - 1/3

Example 3 (continued):

(-3/5)(2) = -6/5 (-3/5)(4/5) = -12/25

So the expression becomes: -6/5 + (-12/25)

Step 4: Simplify and Combine (if necessary)

Finally, simplify the resulting fractions and combine them if needed. This may involve finding a common denominator if you are adding or subtracting fractions.

Example 1 (conclusion):

To add 3/8 and 1/16, we find a common denominator (16):

3/8 = 6/16 6/16 + 1/16 = 7/16

Therefore, (1/2)(3/4 + 1/8) = 7/16

Example 2 (conclusion):

To subtract 1/3 from 5/9, we find a common denominator (9):

1/3 = 3/9 5/9 - 3/9 = 2/9

Therefore, (2/3)(5/6 - 1/2) = 2/9

Example 3 (conclusion):

To add -6/5 and -12/25, we find a common denominator (25):

-6/5 = -30/25 -30/25 + (-12/25) = -42/25

Therefore, (-3/5)(2 + 4/5) = -42/25

Handling Mixed Numbers

When working with mixed numbers, remember to convert them to improper fractions before applying the distributive property. This ensures consistent application of the multiplication rules for fractions.

Example:

(1 1/2)(2/3 + 1/6)

First, convert 1 1/2 to an improper fraction: (3/2)

Now apply the distributive property:

(3/2)(2/3 + 1/6) = (3/2)(2/3) + (3/2)(1/6) = 1 + 1/4 = 5/4

Distributive Property with Variables and Fractions

The distributive property works equally well with algebraic expressions involving variables and fractions.

Example:

(1/3)(6x + 9y) = (1/3)(6x) + (1/3)(9y) = 2x + 3y

Common Mistakes to Avoid

• Forgetting to distribute to every term: Ensure the fraction is multiplied by every term inside the parentheses.
• Incorrect fraction multiplication: Remember to multiply numerators and denominators separately.
• Not simplifying fractions: Always simplify your final answer to its lowest terms.
• Ignoring negative signs: Carefully handle negative signs in both the fraction and the terms within the parentheses.

Practice Problems

To solidify your understanding, try solving these practice problems:

1. (1/4)(8 + 12)
2. (2/5)(15 - 10)
3. (3/7)(14/3 + 7/2)
4. (-1/2)(6x - 4y)
5. (2 1/3)(3/4 + 6/5)

Conclusion

Mastering the distributive property with fractions is a crucial skill for success in algebra and beyond. By following the step-by-step guide, understanding the common pitfalls, and practicing regularly, you can confidently tackle any expression involving fractions and the distributive property. Remember to break down the problem into smaller, manageable steps, and always check your work for accuracy and simplification. With consistent practice, this initially challenging concept will become second nature.