Adding fractions with different denominators requires finding a common denominator before you can perform the addition. Here's a breakdown of the process:
Steps to Add Fractions with Unlike Denominators
- Find a Common Denominator: The denominators of the fractions must be the same before you can add them. The easiest way to find a common denominator is to find the Least Common Multiple (LCM) of the denominators.
- Convert the Fractions: Once you have a common denominator, you need to convert each fraction so that it has this denominator. To do this, multiply both the numerator and denominator of each fraction by the number that makes the original denominator equal to the common denominator.
- Add the Numerators: Now that the fractions have the same denominator, you can add the numerators together. Keep the denominator the same.
- Simplify: Finally, simplify the resulting fraction if possible.
Methods for Finding a Common Denominator and Adding
Here are a couple of methods to tackle this:
Method 1: Least Common Multiple (LCM)
- Identify the Denominators: Determine the denominators of the fractions you want to add.
- Find the LCM: Calculate the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators.
- Convert the Fractions: Convert each fraction to an equivalent fraction with the LCM as the new denominator.
- Add the Numerators: Add the numerators of the fractions, keeping the common denominator.
- Simplify: Simplify the resulting fraction, if possible.
Method 2: Cross Multiplication (As mentioned in the reference)
This method provides a shortcut, especially when dealing with two fractions:
- Cross Multiply: Multiply the numerator of the first fraction by the denominator of the second fraction. Then, multiply the numerator of the second fraction by the denominator of the first fraction. These will be the new numerators.
- Find the Common Denominator: Multiply the two original denominators together. This will be the common denominator.
- Add the New Numerators: Add the results from the cross-multiplication step. Place this sum over the common denominator.
- Simplify: Simplify the resulting fraction if needed.
Example (Using cross multiplication from the reference):
Add 3/8 + 5/7
- Cross multiply: (3 x 7 = 21) and (8 x 5 = 40).
- New Numerators: 21 and 40
- Common Denominator: 8 x 7 = 56
- Add New Numerators: 21 + 40 = 61
- Result: 61/56
Example
Let's add 1/4 + 2/3:
- Common Denominator: The LCM of 4 and 3 is 12.
- Convert Fractions:
- 1/4 = (1 x 3) / (4 x 3) = 3/12
- 2/3 = (2 x 4) / (3 x 4) = 8/12
- Add Numerators: 3/12 + 8/12 = (3 + 8) / 12 = 11/12
- Simplify: 11/12 is already in its simplest form.
Tips for Success
- Always simplify your final answer to its lowest terms.
- Double-check your calculations to avoid errors.
- Practice makes perfect! The more you practice, the easier it will become.
