The distributive property of rational numbers under subtraction states that multiplying a rational number by the difference of two other rational numbers is equivalent to multiplying the first number by each of the other two separately and then subtracting the products. This fundamental algebraic property simplifies calculations and is expressed formally as:
For any rational numbers a, b, and c:
a × (b - c) = (a × b) - (a × c)
This property ensures consistency in arithmetic operations involving rational numbers.
Understanding the Distributive Property
The distributive property is a core concept in algebra that connects the operations of multiplication and subtraction. It essentially allows you to "distribute" a factor (the first number) to each term inside a parenthesis where subtraction is involved.
Key Aspects:
- Rational Numbers: These are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Examples include 1/2, -3/4, 5 (which can be written as 5/1), and 0.25 (which is 1/4). You can learn more about rational numbers on Wikipedia.
- Subtraction: The property applies when one rational number is multiplied by the difference between two other rational numbers.
- Order of Operations: It provides an alternative, and often simpler, way to solve expressions by breaking them down. Instead of first performing the subtraction inside the parenthesis, you can distribute the multiplication first.
How It Works: The Breakdown
Consider the expression a × (b - c).
- Direct Calculation: First, you would calculate the difference (b - c), and then multiply the result by a.
- Distributive Method: Alternatively, you can multiply a by b to get ab, then multiply a by c to get ac, and finally subtract ac from ab.
Both methods yield the same result, confirming the property. This property is crucial for algebraic manipulation and solving equations. For additional insights into the distributive property, explore resources like Khan Academy's explanation.
Practical Examples
Let's illustrate the distributive property with various rational numbers.
Example 1: Using Fractions
Let a = 2/3, b = 1/2, and c = 1/4.
Applying a × (b - c):
- 2/3 × (1/2 - 1/4)
- 2/3 × (2/4 - 1/4)
- 2/3 × (1/4)
- 2/12 = 1/6
Applying (a × b) - (a × c):
- (2/3 × 1/2) - (2/3 × 1/4)
- (2/6) - (2/12)
- (1/3) - (1/6)
- (2/6) - (1/6)
- 1/6
As demonstrated, both approaches result in 1/6.
Example 2: Using Decimals (also rational numbers)
Let a = 0.5, b = 10, and c = 4.
Applying a × (b - c):
- 0.5 × (10 - 4)
- 0.5 × (6)
- 3
Applying (a × b) - (a × c):
- (0.5 × 10) - (0.5 × 4)
- (5) - (2)
- 3
Again, the results match.
Example 3: Negative Rational Numbers
Let a = -1/2, b = 6, and c = 2.
Applying a × (b - c):
- -1/2 × (6 - 2)
- -1/2 × (4)
- -2
Applying (a × b) - (a × c):
- (-1/2 × 6) - (-1/2 × 2)
- (-3) - (-1)
- -3 + 1
- -2
Summary Table
This table summarizes the property with a general example:
| Expression Format | Calculation Steps | Final Result |
|---|---|---|
a × (b - c) |
1. Subtract c from b. |
R |
2. Multiply a by the difference. |
||
(a × b) - (a × c) |
1. Multiply a by b. |
R |
2. Multiply a by c. |
||
| 3. Subtract the second product from the first. |
The distributive property is a powerful tool that simplifies complex expressions involving rational numbers, making arithmetic and algebraic problem-solving more efficient.
