Comparing fractions involves determining which fraction represents a larger or smaller value. Several methods can be used, depending on the fractions being compared. Remember that the numerator is the top number and the denominator is the bottom number.
Comparing Fractions with the Same Denominator
When fractions have the same denominator, comparing them is straightforward.
-
The fraction with the larger numerator is the larger fraction.
- Example: 3/5 > 1/5 because 3 is greater than 1.
-
This works because the denominator indicates the size of each piece, so if the pieces are the same size, then the fraction with more pieces (larger numerator) is the greater fraction.
Comparing Fractions with the Same Numerator
When fractions have the same numerator, the fraction with the smaller denominator is the larger fraction.
- Example: 2/3 > 2/5 because 3 is smaller than 5.
- This is because the denominator indicates how many total pieces the whole is divided into. So a smaller denominator means there are fewer pieces, meaning each piece must be larger.
Comparing Fractions with Different Numerators and Denominators
When fractions have different numerators and denominators, you have two main options:
-
Find a Common Denominator:
- Find the least common multiple (LCM) of the denominators. This becomes the new common denominator.
- Convert each fraction into an equivalent fraction with the common denominator.
- Compare the numerators. The fraction with the larger numerator is the larger fraction.
- Example: Comparing 1/3 and 2/5.
- The LCM of 3 and 5 is 15.
- 1/3 = 5/15 (multiply numerator and denominator by 5).
- 2/5 = 6/15 (multiply numerator and denominator by 3).
- Since 6/15 > 5/15, then 2/5 > 1/3.
-
Cross-Multiplication:
- Multiply the numerator of the first fraction by the denominator of the second fraction.
- Multiply the numerator of the second fraction by the denominator of the first fraction.
- Compare the two products.
- If the first product is larger, the first fraction is larger. If the second product is larger, the second fraction is larger. If the products are equal, the fractions are equal.
- Example: Comparing 3/4 and 5/7.
- 3 * 7 = 21
- 5 * 4 = 20
- Since 21 > 20, then 3/4 > 5/7.
Summary Table
| Condition | Rule | Example |
|---|---|---|
| Same Denominator | Larger numerator means larger fraction. | 3/7 > 2/7 |
| Same Numerator | Smaller denominator means larger fraction. | 4/5 > 4/9 |
| Different Numerators & Denominators | Find a common denominator or use cross-multiplication. | Compare 1/2 & 2/3 |
By understanding these methods, you can easily compare any fractions.
