Mathematical Equivalence: a ÷ b × c a × c ÷ b

Understanding the principles of mathematical equivalence is essential for mastering advanced algebraic concepts. One of the key properties in mathematics is the distributive property, which allows us to manipulate expressions in a way that preserves their value. The expression a ÷ b × c is equivalent to a × c ÷ b, provided that b ≠ 0. This property is not only useful in simplifying expressions but also in solving complex equations.

Explanation of the Distributive Property

The distributive property is a fundamental principle in mathematics that allows us to distribute multiplication over addition or subtraction. Similarly, in our case, the distributive property can be applied to multiplication and division. When we see the expression a ÷ b × c, we can understand it as performing the division first, which can be rewritten as (a ÷ b) × c. However, by the distributive property, we can also rewrite it as a × (c ÷ b), which is equivalent to a × c ÷ b.

Example and Practical Usage

Let's consider an example to illustrate the concept more clearly. Suppose we have the expression 8 ÷ 2 × 3. According to the principles of multiplication and division, we can perform the operations in any order. First, we can divide 8 by 2, which gives us 4. Then, we multiply the result by 3, giving us 12. Alternatively, we can rewrite the expression as 8 × (3 ÷ 2), which simplifies to 8 × 1.5, again resulting in 12. Both methods yield the same result, confirming the mathematical equivalence.

Applying the Property in Algebraic Expressions

This property is particularly valuable when simplifying complex algebraic expressions. For instance, consider the expression 5x ÷ 2y × 4z. Using the distributive property, we can rewrite it as 5x × (4z ÷ 2y). This simplifies further to 5x × 2z ÷ y, which is much easier to work with in further calculations or equations.

Order of Operations

Understanding and correctly applying the order of operations is crucial for solving mathematical expressions. In the expression a ÷ b × c, the order of operations must be considered, as the result is affected by the sequence of operations. According to the standard order of operations (PEMDAS/BODMAS), division and multiplication have the same precedence, and should be performed from left to right.

Conclusion

The property that a ÷ b × c a × c ÷ b is a powerful tool in algebra and is widely utilized in simplifying expressions and solving equations. By mastering this concept and the distributive property, students and professionals in mathematics and related fields can enhance their problem-solving skills and work more efficiently.

References

[1] Math is Fun - Order of Operations
[2] Wikipedia - Division (Mathematics)