To find common fractions with different denominators, you need to find a common denominator, which allows you to compare, add, or subtract fractions easily. Here's a breakdown of how to achieve this, drawing from the provided reference:
Understanding the Need for Common Denominators
Fractions with different denominators (unlike denominators) cannot be directly added or compared. A common denominator provides a shared unit, allowing us to perform arithmetic operations.
- Equivalent Fractions: The key to finding a common denominator is to create equivalent fractions that represent the same value, but have the desired common denominator.
Finding the Common Denominator
There are several ways to find a common denominator, but one of the most common and efficient method is to find the Least Common Multiple (LCM) of the denominators. The LCM of the denominators is used as the common denominator.
- The Least Common Multiple (LCM): The smallest number that is a multiple of all the denominators.
Steps to Find Common Fractions
Here are the steps to find common fractions with different denominators:
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Identify the Denominators: Determine the denominators of the fractions you want to work with. For example, if you have 1/3 and 2/9, the denominators are 3 and 9.
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Find the Least Common Multiple (LCM): Find the LCM of the denominators. In the example above, the LCM of 3 and 9 is 9.
- The multiple of 3 are 3, 6, 9, 12...
- The multiple of 9 are 9, 18, 27...
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Convert to Equivalent Fractions: Convert each fraction to an equivalent fraction that has the LCM as the new denominator.
- For 1/3: To convert 1/3 to an equivalent fraction with a denominator of 9, multiply both the numerator and the denominator by 3 (9 / 3 = 3): (1 x 3) / (3 x 3) = 3/9.
- For 2/9: The fraction 2/9 already has a denominator of 9, so no change is needed.
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Result: Now you have fractions with a common denominator. So, 1/3 is equivalent to 3/9, and 2/9 remain the same and they can be easily added, subtracted or compared.
- The reference illustrates how 3/9 is equivalent to 1/3, highlighting the core process of finding a common denominator for adding fractions.
Example
Let's say you want to add 1/3 and 2/9.
- Original fractions: 1/3 and 2/9
- Common denominator: The LCM of 3 and 9 is 9.
- Equivalent fractions: 1/3 becomes 3/9, and 2/9 remains 2/9.
- Now you can add them: 3/9 + 2/9 = 5/9
Conclusion
Finding common fractions with different denominators involves finding a common denominator, usually the LCM, and converting each fraction to an equivalent fraction with that denominator. This process enables you to perform arithmetic operations, and it allows for easy comparisons.
