An equivalent fraction by division is a fraction that represents the same value as another fraction but is expressed in simpler or lower terms. This is achieved by dividing both the numerator (top number) and the denominator (bottom number) of the original fraction by the same non-zero number, which must be a common factor of both.
What is an Equivalent Fraction by Division?
An equivalent fraction by division is a fraction derived from an original fraction by simplifying it. This process, also known as reducing a fraction or simplifying to lowest terms, results in a new fraction that has the identical value as the original, even though its numbers are smaller. The core principle is to maintain the ratio between the numerator and the denominator, ensuring the fraction still represents the same part of a whole.
How Division Creates Equivalent Fractions
To generate an equivalent fraction through division, you must find a common factor—a number that divides evenly into both the numerator and the denominator. By dividing both parts by this common factor, you effectively express the fraction in a simpler form without altering its fundamental value.
This method is crucial because it reduces fractions to their simplest form, making them easier to understand, compare, and use in calculations. When fractions are simplified this way, they are reduced to the same value or number, demonstrating their equivalence.
Steps to Find an Equivalent Fraction by Division
- Identify the fraction: Start with the fraction you want to simplify.
- Find a common factor: Determine a number (greater than 1) that can divide both the numerator and the denominator without leaving a remainder.
- Divide: Divide both the numerator and the denominator by this common factor.
- Repeat (if necessary): If the new fraction still has common factors (other than 1), repeat steps 2 and 3 until no more common factors exist. The fraction is then in its simplest form, also known as its lowest terms.
Examples of Equivalent Fractions by Division
Here are some practical examples demonstrating how division is used to find equivalent fractions:
| Original Fraction | Common Factor | Division Performed | Equivalent Fraction |
|---|---|---|---|
| 8/12 | 4 | (8 ÷ 4) / (12 ÷ 4) | 2/3 |
| 10/20 | 10 | (10 ÷ 10) / (20 ÷ 10) | 1/2 |
| 15/25 | 5 | (15 ÷ 5) / (25 ÷ 5) | 3/5 |
| 12/18 | 2, then 3 (or 6) | (12 ÷ 2) / (18 ÷ 2) = 6/9 (6 ÷ 3) / (9 ÷ 3) = 2/3 |
2/3 |
In the last example (12/18), you could have divided by the greatest common factor (GCF) directly, which is 6, to get to 2/3 in one step.
Why Simplify Fractions?
Simplifying fractions by division serves several important purposes in mathematics and everyday life:
- Clarity and Simplicity: Reduced fractions are easier to comprehend at a glance. For instance, "1/2" is intuitively simpler than "50/100" to represent half.
- Easier Comparisons: Comparing fractions is much simpler when they are in their lowest terms. It's easier to see that 2/3 is larger than 1/2 than to compare 12/18 and 5/10.
- Standardization: Often, answers to mathematical problems are expected to be in simplest form. This ensures consistency across different calculations.
- Foundation for Complex Operations: Working with simplified fractions makes subsequent arithmetic operations (like addition, subtraction, multiplication, and division) less cumbersome.
Understanding equivalent fractions, particularly through division, is a fundamental concept in mathematics that underpins various calculations and problem-solving techniques. It highlights that numbers can be represented in different ways while maintaining the same value, a core principle of numerical fluency. For more information on fractions and their properties, you can explore resources on equivalent fractions or simplifying fractions.
