To find the bigger fraction, you need a method to compare their values accurately. The most common and reliable techniques involve finding a common denominator, using cross-multiplication, or converting fractions to decimals.
Key Methods to Compare Fractions
Comparing fractions effectively helps you understand their relative sizes. Here are the most straightforward methods:
1. Finding a Common Denominator
This method converts fractions into equivalent forms that share the same denominator, making direct comparison of their numerators possible.
How it Works:
- Find the Least Common Multiple (LCM) of the Denominators: The LCM will be your new common denominator. This is the smallest number that both denominators can divide into evenly.
- Convert Each Fraction: For each fraction, determine what you multiplied its original denominator by to get the LCM. Then, multiply its numerator by the same number to create an equivalent fraction.
- Compare the Numerators: Once both fractions have the same denominator, the fraction with the larger numerator is the bigger fraction.
Example: Comparing 3/4 and 5/6
Let's determine which fraction is larger: 3/4 or 5/6.
- Step 1: Find the LCM of 4 and 6.
- Multiples of 4: 4, 8, 12, 16, 20...
- Multiples of 6: 6, 12, 18, 24...
- The LCM of 4 and 6 is 12.
- Step 2: Convert the fractions to have a denominator of 12.
- For 3/4: To get a denominator of 12, multiply 4 by 3. So, multiply the numerator (3) by 3 as well:
(3 * 3) / (4 * 3) = 9/12. - For 5/6: To get a denominator of 12, multiply 6 by 2. So, multiply the numerator (5) by 2 as well:
(5 * 2) / (6 * 2) = 10/12.
- For 3/4: To get a denominator of 12, multiply 4 by 3. So, multiply the numerator (3) by 3 as well:
- Step 3: Compare the new numerators.
- Now you are comparing 9/12 and 10/12.
- Since 10 is greater than 9, 10/12 is larger than 9/12.
Therefore, 5/6 is larger than 3/4.
Summary Table:
| Original Fraction | Denominator | Multiplier to Reach LCM (12) | Equivalent Fraction |
|---|---|---|---|
| 3/4 | 4 | 3 | 9/12 |
| 5/6 | 6 | 2 | 10/12 |
2. Cross-Multiplication
Cross-multiplication is a quick method that doesn't require finding a common denominator explicitly.
How it Works:
- 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 Products: The fraction corresponding to the larger product is the bigger fraction.
Example: Comparing 2/3 and 4/5
- Step 1: Multiply the numerator of the first fraction (2) by the denominator of the second fraction (5):
2 * 5 = 10. Write this product above the first fraction. - Step 2: Multiply the numerator of the second fraction (4) by the denominator of the first fraction (3):
4 * 3 = 12. Write this product above the second fraction. - Step 3: Compare the products: 10 vs. 12. Since 12 is greater than 10, the second fraction (4/5) is larger.
Therefore, 4/5 is larger than 2/3.
3. Converting to Decimals
This method is straightforward if you're comfortable with division.
How it Works:
- Divide the numerator by the denominator for each fraction.
- Compare the decimal values. The larger decimal corresponds to the bigger fraction.
Example: Comparing 7/8 and 5/6
- For 7/8:
7 ÷ 8 = 0.875 - For 5/6:
5 ÷ 6 ≈ 0.833
Since 0.875 is greater than 0.833, 7/8 is larger than 5/6.
4. Visual Comparison or Estimation
For simple fractions, you can often make a quick comparison by visualizing or estimating their values relative to benchmarks like 0, 1/2, or 1.
Tips for Estimation:
- Compare to 1/2: Is the fraction greater than, less than, or equal to 1/2? (e.g., 2/3 is greater than 1/2, 1/4 is less than 1/2).
- Compare to 1: How close is the numerator to the denominator? (e.g., 7/8 is very close to 1, while 1/8 is not).
- Same Numerator: If fractions have the same numerator, the fraction with the smaller denominator is larger (e.g., 1/2 > 1/3 because a whole is divided into fewer, larger pieces).
For more complex comparisons, one of the first three precise methods is always recommended.
When to Use Each Method
- Common Denominator: Ideal when denominators are small and their LCM is easy to find. Good for understanding fraction equivalence.
- Cross-Multiplication: Excellent for quick comparisons without needing the exact common denominator, especially with larger numbers.
- Converting to Decimals: Best when you need an approximate value or when calculators are readily available.
Understanding these methods empowers you to confidently determine the bigger fraction in any scenario.
