Add, subtract, multiply, divide fractions with step-by-step solutions. Mixed numbers, simplify to lowest terms, and compare — all in one tool.
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The fundamental rule of fraction addition is that you can only add fractions that share the same denominator. When denominators differ, you must find the Least Common Denominator (LCD) — the smallest number that both denominators divide into evenly — convert both fractions to equivalent forms using that LCD, then add the numerators while keeping the denominator. For 1/2 + 1/3: the LCD of 2 and 3 is 6. Convert: 1/2 = 3/6 and 1/3 = 2/6. Now add: 3/6 + 2/6 = 5/6. The LCD equals the Least Common Multiple (LCM) of the denominators, which you can find by listing multiples or using the formula LCD = (a × b) / GCD(a, b).
Subtraction follows the identical process — find the LCD, convert to equivalent fractions, then subtract the numerators. The crucial detail is tracking signs: when subtracting a negative numerator, the double negative becomes addition. For 3/4 − 1/6: LCD = 12, so 9/12 − 2/12 = 7/12. After computing the result, always check whether the answer simplifies — divide both numerator and denominator by their Greatest Common Divisor (GCD). If the GCD is 1, the fraction is already in lowest terms. If not, divide both by the GCD to produce the simplified form.
Multiplication is actually the simplest fraction operation — no LCD needed. Simply multiply numerators together and denominators together: (a/b) × (c/d) = (a×c) / (b×d). For 2/3 × 3/4: numerators give 2×3=6, denominators give 3×4=12, so the result is 6/12, which simplifies to 1/2. A powerful shortcut is cross-cancellation before multiplying: if any numerator shares a common factor with any denominator (not necessarily the paired one), cancel that factor first. This keeps numbers smaller and often eliminates the need to simplify afterward. For 6/8 × 4/9: 6 and 9 share a factor of 3 (giving 2 and 3), and 8 and 4 share a factor of 4 (giving 2 and 1). Result: (2×1)/(2×3) = 2/6 = 1/3.
Division uses the "Keep-Change-Flip" (KCF) method, also called multiply by the reciprocal. Keep the first fraction unchanged, change the division sign to multiplication, and flip (take the reciprocal of) the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). For 1/2 ÷ 1/4: keep 1/2, change ÷ to ×, flip 1/4 to 4/1, multiply: 1/2 × 4/1 = 4/2 = 2. This works because dividing by a fraction is the same as multiplying by its reciprocal — the two operations are mathematical inverses of each other.
A mixed number combines a whole number and a proper fraction — like 2½ or 3¾. Before performing any arithmetic with mixed numbers, convert them to improper fractions (where numerator ≥ denominator). The conversion formula: multiply the whole number by the denominator, add the numerator, place the result over the original denominator. For 2½: (2×2) + 1 = 5, over denominator 2, giving 5/2. For 3¾: (3×4) + 3 = 15, over 4, giving 15/4. Once converted, apply the standard operation (add, subtract, multiply, or divide) and convert the answer back to a mixed number by dividing: quotient becomes the whole, remainder becomes the new numerator over the original denominator.
Converting an improper fraction back to a mixed number is simple division with remainder. For 17/5: 17 ÷ 5 = 3 with remainder 2, so the mixed number is 3⅖. For 22/7: 22 ÷ 7 = 3 remainder 1, giving 3 1/7. This back-and-forth conversion is essential for mixed number arithmetic. Note that you should simplify the improper fraction before converting back — 20/8 simplifies to 5/2, which then converts to 2½, cleaner than converting 20/8 directly to 2 4/8 and then simplifying the fractional part separately.
A fraction is in lowest terms (simplest form) when the numerator and denominator share no common factor other than 1 — their Greatest Common Divisor is 1. To simplify, find the GCD of numerator and denominator, then divide both by it. For 36/48: find GCD(36,48). Using prime factorization: 36 = 2² × 3² and 48 = 2⁴ × 3. The GCD is 2² × 3 = 12. Divide: 36÷12 = 3 and 48÷12 = 4. Simplified: 3/4. The Euclidean algorithm provides a faster GCD calculation for large numbers: repeatedly replace the larger number with the remainder of dividing larger by smaller until the remainder is 0. The last non-zero remainder is the GCD.
A practical simplification shortcut: if both numbers are even, divide by 2 repeatedly until at least one is odd. Then check divisibility by 3 (digits sum divisible by 3), 5 (ends in 0 or 5), 7, 11, and so on. Each time you find a common factor, divide both by it. Continue until no common factors remain. For 72/90: both even, divide by 2 → 36/45. Now 36 and 45 are both divisible by 9 (36=9×4, 45=9×5) → 4/5. GCD(4,5) = 1, so 4/5 is fully simplified. This step-by-step approach is often faster mentally than computing the GCD directly for numbers you encounter in everyday arithmetic.
The most reliable method to compare two fractions is cross-multiplication. To compare a/b vs c/d, compute a×d and b×c. If a×d > b×c, then a/b > c/d. If a×d < b×c, then a/b < c/d. If equal, the fractions are equivalent. For 2/3 vs 3/4: cross-multiply to get 2×4=8 and 3×3=9. Since 8 < 9, we have 2/3 < 3/4. This works because converting both fractions to the common denominator b×d and comparing numerators gives exactly these cross products. Cross-multiplication is the butterfly method — draw an X between the fractions and compare the products at the top of each wing.
The decimal method is often faster for intuitive comparison: divide each fraction's numerator by its denominator. 2/3 ≈ 0.667 and 3/4 = 0.75. Since 0.667 < 0.75, clearly 2/3 < 3/4. For ranking multiple fractions, convert all to decimals, rank the decimals, then map back to the original fractions. This approach scales well to comparing three or more fractions simultaneously. The LCD method (convert all fractions to the same denominator and compare numerators) is most useful in teaching contexts because it makes the comparison visual and explicit, but cross-multiplication is fastest for quick two-fraction comparisons.