An equivalent fraction for 1/2, specifically "example 10" derived by multiplying both the numerator and denominator by 10, is 10/20.
Understanding Equivalent Fractions
Equivalent fractions are different fractions that represent the same value or proportion of a whole. They may look different numerically, but they signify the identical quantity. For instance, 1/2, 2/4, 3/6, and 4/8 are all equivalent fractions. Their values are equal, meaning they represent the same portion of a whole. Imagine these fractions represented as circles with shaded parts; it can be seen that the shaded parts in all the figures represent the same portion if seen as a whole. This concept is fundamental in understanding how fractions work.
Identifying "Example 10": The Equivalent of 1/2 as 10/20
When discussing "example 10" in the context of equivalent fractions, it commonly refers to generating an equivalent fraction by multiplying the numerator and denominator of a base fraction by the factor of 10. Using the well-known fraction 1/2 as our base, the calculation proceeds as follows:
- Original Fraction: 1/2
- Multiply by 10/10 (which equals 1, so the value doesn't change): (1 × 10) / (2 × 10)
- Resulting Equivalent Fraction: 10/20
Therefore, 10/20 is an equivalent fraction of 1/2, representing "example 10" by virtue of the multiplier used. Both 1/2 and 10/20 occupy the exact same point on a number line, confirming their identical value.
How to Find Equivalent Fractions
Finding equivalent fractions is a straightforward process involving multiplication or division:
- Multiplication: Multiply both the numerator (top number) and the denominator (bottom number) of a fraction by the same non-zero whole number.
- Example: To find an equivalent fraction for 2/3, multiply both by 2:
(2 × 2) / (3 × 2) = 4/6 - Multiply by 3: (2 × 3) / (3 × 3) = 6/9
- Example: To find an equivalent fraction for 2/3, multiply both by 2:
- Division (Simplifying): Divide both the numerator and the denominator by their greatest common divisor (GCD). This process is also known as simplifying or reducing a fraction to its lowest terms.
- Example: To simplify 15/25, the GCD of 15 and 25 is 5.
(15 ÷ 5) / (25 ÷ 5) = 3/5
- Example: To simplify 15/25, the GCD of 15 and 25 is 5.
More Examples of Equivalent Fractions
Here are additional examples illustrating various equivalent fraction pairs:
- 1/3 = 2/6 = 3/9
- 2/5 = 4/10 = 6/15
- 3/4 = 6/8 = 9/12
- 5/10 = 1/2 = 25/50
Table of Equivalent Fractions and Their Decimal Values:
| Original Fraction | Multiplier | Equivalent Fraction | Decimal Value |
|---|---|---|---|
| 1/2 | - | 1/2 | 0.5 |
| 1/2 | 2/2 | 2/4 | 0.5 |
| 1/2 | 3/3 | 3/6 | 0.5 |
| 1/2 | 10/10 | 10/20 | 0.5 |
| 2/3 | - | 2/3 | 0.666... |
| 2/3 | 2/2 | 4/6 | 0.666... |
| 3/4 | - | 3/4 | 0.75 |
| 3/4 | 5/5 | 15/20 | 0.75 |
Visualizing Equivalent Fractions
Visual aids, such as pie charts or fraction bars, are excellent tools for understanding equivalent fractions. When 1/2, 2/4, 3/6, and 10/20 are visually represented, they all show the same amount of shaded area, demonstrating that they are indeed equal in value. This visual confirmation reinforces the mathematical concept that different fractional representations can denote the same quantity.
Importance and Real-World Applications
Understanding equivalent fractions is crucial for various mathematical operations, including adding or subtracting fractions with different denominators, comparing fractions, and simplifying complex expressions. Beyond the classroom, equivalent fractions are used in everyday situations like:
- Cooking: Adjusting recipes that call for 1/2 cup of an ingredient if you only have a 1/4 cup measure (which you'd use twice, making 2/4).
- Construction: Measuring and cutting materials accurately, where 1/2 inch is the same as 4/8 inch.
- Finances: Comparing parts of a budget or ownership stakes.
Knowing how to identify and generate equivalent fractions simplifies many mathematical problems and provides a deeper understanding of fractional quantities.
