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To add fractions with different denominators, you must first rewrite them as equivalent fractions with a common denominator. 1/3 + 1/4: the least common denominator of 3 and 4 is 12, so 1/3 = 4/12 and 1/4 = 3/12, giving 4/12 + 3/12 = 7/12. The key insight is that you cannot add fractions measured in different-sized units -- thirds and fourths are different units, just as you cannot add 3 feet and 4 inches without converting. Finding the least common multiple of the denominators gives the most efficient common denominator, though any common multiple works.
Use visual models first: overlay fraction strips of thirds and fourths to see that twelfths is the common subdivision. Practice finding least common multiples before applying to fraction addition. Work through many examples with different denominator pairs. Simplify results. Extend to mixed numbers once fluent with proper fractions.
You already know how to add fractions with the same denominator: 3/8 + 2/8 = 5/8. The denominator names what kind of piece you have — eighths — and you simply count them. The challenge with unlike denominators is that you're combining pieces of different sizes. Trying to add thirds and fourths directly is like trying to add 3 feet and 4 inches: the units don't match, so you can't just count.
The fix is to rewrite both fractions so they use the same-sized pieces. For 1/3 + 1/4, you need a number that is both a multiple of 3 and a multiple of 4. List multiples of each: multiples of 3 are 3, 6, 9, 12, 15...; multiples of 4 are 4, 8, 12, 16... The least common multiple is 12, so use 12 as your common denominator.
Now apply equivalent fractions. To convert 1/3 to twelfths, ask: 3 × ? = 12? The answer is 4, so multiply numerator and denominator by 4: 1/3 = 4/12. For 1/4: 4 × ? = 12? Multiply by 3: 1/4 = 3/12. Now add: 4/12 + 3/12 = 7/12. The pieces are the same size, so the numerators can be added directly.
A shortcut that always works: multiply the two denominators to get a common denominator. For 1/3 + 1/4, use 3 × 4 = 12, which happens to be the LCD here. But for 1/4 + 1/6, the product is 24, while the LCD is only 12 — using 12 keeps numbers smaller and often avoids simplifying at the end. Either approach gives the correct answer; the LCD is more efficient.
The most important thing to avoid: never add the denominators. The answer to 1/3 + 1/4 is not 2/7. Imagine eating one slice from a 3-slice pizza and one slice from a 4-slice pizza. You now have 2 slices, but you don't have 2 slices of a 7-slice pizza — the pieces are different sizes. The denominator describes the size of each piece; it is not a number to be added.