The principle of equality of fractions states that a fraction remains equivalent to its original value if both its numerator and denominator are multiplied or divided by the same non-zero number.
This principle is fundamental in simplifying fractions, comparing fractions, and performing arithmetic operations with fractions. It allows us to manipulate fractions without changing their inherent value.
Explanation
The principle can be mathematically expressed as follows:
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For Multiplication: a/b = (a n) / (b n), where 'a' is the numerator, 'b' is the denominator, and 'n' is any non-zero number.
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For Division: a/b = (a / n) / (b / n), where 'a' is the numerator, 'b' is the denominator, and 'n' is any non-zero number that divides both 'a' and 'b' evenly.
In simpler terms, if you multiply or divide both the top (numerator) and bottom (denominator) of a fraction by the same number, you get a new fraction that represents the exact same value as the original one. This works because you're essentially multiplying or dividing the fraction by 1 (in the form of n/n), which doesn't change its value.
Examples
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Multiplication Example:
Consider the fraction 1/2. If we multiply both the numerator and denominator by 3, we get:
(1 3) / (2 3) = 3/6
1/2 and 3/6 are equivalent fractions; they both represent the same value (0.5).
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Division Example:
Consider the fraction 4/8. If we divide both the numerator and denominator by 4, we get:
(4 / 4) / (8 / 4) = 1/2
4/8 and 1/2 are equivalent fractions; they both represent the same value (0.5).
Why is this principle important?
- Simplifying Fractions: We can use this principle to reduce fractions to their simplest form (also known as lowest terms) by dividing the numerator and denominator by their greatest common divisor (GCD).
- Finding Equivalent Fractions: We can create fractions with a common denominator, which is crucial for adding and subtracting fractions.
- Comparing Fractions: By expressing fractions with a common denominator using this principle, we can easily compare their values.
In Summary
The principle of equality of fractions is a cornerstone of fraction manipulation, allowing us to create equivalent forms of a fraction without altering its intrinsic value. This is achieved by multiplying or dividing both the numerator and denominator by the same non-zero number.
