Subtracting mixed fractions with different denominators involves several key steps to ensure accurate calculations. Here’s a step-by-step guide:
1. Convert Mixed Fractions to Improper Fractions
The first step is to convert each mixed fraction into an improper fraction. To do this, multiply the whole number part by the denominator and add the numerator. Keep the same denominator.
-
Formula: Whole Number + (Numerator / Denominator) becomes ((Whole Number * Denominator) + Numerator) / Denominator
-
Example: Convert 2 1/3 to an improper fraction:
- (2 * 3) + 1 = 7
- So, 2 1/3 = 7/3
2. Find a Common Denominator
If the fractions have different denominators, you need to find the least common denominator (LCD). This is the smallest number that both denominators can divide into evenly.
-
How to find the LCD: List the multiples of each denominator until you find a common multiple. The smallest one is the LCD.
-
Example: Subtract 7/3 - 5/2. The denominators are 3 and 2.
- Multiples of 3: 3, 6, 9, 12...
- Multiples of 2: 2, 4, 6, 8...
- The LCD is 6.
3. Convert Fractions to Equivalent Fractions with the Common Denominator
Now, convert each fraction to an equivalent fraction using the LCD. To do this, determine what number you need to multiply the original denominator by to get the LCD. Then, multiply both the numerator and the denominator by that number.
- Example (Continuing from above):
- To convert 7/3 to a fraction with a denominator of 6, multiply both the numerator and denominator by 2: (7 2) / (3 2) = 14/6
- To convert 5/2 to a fraction with a denominator of 6, multiply both the numerator and denominator by 3: (5 3) / (2 3) = 15/6
4. Subtract the Fractions
Now that both fractions have the same denominator, you can subtract the numerators. Keep the denominator the same.
- Example: 14/6 - 15/6 = (14 - 15) / 6 = -1/6
5. Simplify the Result (if possible)
If the resulting fraction is improper (numerator is greater than or equal to the denominator), convert it back to a mixed number. Also, simplify the fraction if possible by dividing both the numerator and denominator by their greatest common factor (GCF).
- Example: Our result is -1/6. This is already a simplified proper fraction (the absolute value of the numerator is smaller than the denominator, and there is no common factor between the numerator and the denominator besides 1), so no further simplification is needed. If we had gotten 4/8, we could divide both by 4 to get 1/2.
6. Consider Borrowing (If necessary during subtraction)
If, after finding a common denominator, the fraction you are subtracting from has a smaller numerator than the fraction you are subtracting, you'll need to "borrow" from the whole number part (if there is one) to increase the value of the numerator.
-
Example: Let's say you are trying to subtract 3 2/5 - 1 4/5. Convert to improper fractions: 17/5 - 9/5. This is a straightforward subtraction, and yields 8/5.
-
Example (Borrowing): Let's say you are trying to subtract 3 1/5 - 1 4/5. After finding common denominators (which in this case are already the same), you have 3 1/5 - 1 4/5. Since 1/5 is less than 4/5, you need to borrow. Borrow 1 from the 3, making it 2. That 1 you borrowed is equal to 5/5. Add that to the existing 1/5: 5/5 + 1/5 = 6/5. Now your problem is 2 6/5 - 1 4/5. Converting to improper fractions, you have 16/5 - 9/5, which yields 7/5.
By following these steps, you can accurately subtract mixed fractions with different denominators.
