Adding and Subtracting Fractions with Like Denominators
When two fractions share the same denominator, the arithmetic becomes straightforward. You only manipulate the numerators; the denominator stays put.
For example, 3/8 + 2/8 simply becomes (3 + 2)/8 = 5/8. The same applies to subtraction: 7/9 − 4/9 = (7 − 4)/9 = 3/9, which simplifies to 1/3.
The key principle: combine numerators while keeping the denominator unchanged. This is where fraction math mirrors integer addition—a rare and welcome simplification. After combining, always reduce your answer to lowest terms by dividing both numerator and denominator by their greatest common divisor (GCD).
Adding and Subtracting Fractions with Unlike Denominators
Fractions with different denominators cannot be added or subtracted directly. You must first rewrite them using a common denominator.
The process:
- Identify the least common multiple (LCM) of both denominators. This becomes your least common denominator (LCD).
- Convert each fraction by multiplying its numerator and denominator by whatever factor makes the denominator equal to the LCD.
- Once both fractions share a denominator, add or subtract the numerators as you would with like denominators.
- Simplify the result if possible.
For instance, to subtract 1/4 from 5/6: the LCM of 4 and 6 is 12. Rewrite 5/6 as 10/12 and 1/4 as 3/12, then compute 10/12 − 3/12 = 7/12.
Working with Mixed Numbers
A mixed number combines a whole number with a fraction, such as 2¾. To add or subtract mixed numbers, convert each to an improper fraction first.
Conversion method: multiply the whole number by the denominator, add the numerator, and place that result over the original denominator. So 2¾ becomes (2 × 4 + 3)/4 = 11/4.
Once both numbers are improper fractions, apply the same rules as before—find a common denominator if needed, then add or subtract. If your final answer is an improper fraction and you want it as a mixed number again, divide the numerator by the denominator: the quotient is the whole part, and the remainder becomes the new numerator.
Core Formula for Fraction Operations
The arithmetic for fractions with a common denominator is immediate. For unlike denominators, you must first align them. Below is the general approach:
For fractions a/b and c/d:
Step 1: Find LCD = LCM(b, d)
Step 2: a/b = (a × (LCD/b)) / LCD
Step 3: c/d = (c × (LCD/d)) / LCD
Step 4: Result = (a × (LCD/b) ± c × (LCD/d)) / LCD
a, c— Numerators of the first and second fractionb, d— Denominators of the first and second fractionLCD— Least common denominator (the LCM of b and d)
Common Pitfalls and Best Practices
Avoid these frequent mistakes when adding or subtracting fractions:
- Don't add or subtract denominators — A common error is treating denominators like numerators. When you add 1/3 + 1/4, the answer is NOT 2/7. You must find the LCD (12 in this case), rewrite as 4/12 + 3/12, then get 7/12.
- Always simplify your final answer — Many calculators and teachers expect answers in lowest terms. After completing the operation, divide both numerator and denominator by their GCD. For example, 6/9 should be reduced to 2/3 before you submit or record your result.
- Convert mixed numbers before operating — It's easy to forget this step. If you're adding 1½ + 2⅓, convert both to improper fractions (3/2 and 7/3) before finding the LCD and adding. Attempting arithmetic on mixed numbers directly often leads to errors.
- Check your LCD carefully — Choosing an incorrect least common denominator wastes time and introduces errors. If denominators are coprime (share no common factors), their LCD is simply their product. For others, use prime factorization or a systematic LCM method.