A fraction is a division waiting to happen: ab\dfrac{a}{b} equals a÷ba \div b, with b≠0b \neq 0. Every rule below follows from that idea and from the fact that multiplying the numerator and denominator by the same number does not change the fraction.

Adding and subtracting

Fractions add directly only when they share a denominator:

ad+bd=a+bd\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}

With different denominators, rewrite both over a common denominator. The smallest one is the least common multiple (LCM).

Worked example: 23+14\dfrac{2}{3} + \dfrac{1}{4}

  1. lcm(3,4)=12\text{lcm}(3, 4) = 12.
  2. 23=812\dfrac{2}{3} = \dfrac{8}{12} and 14=312\dfrac{1}{4} = \dfrac{3}{12}.
  3. Sum: 8+312=1112\dfrac{8 + 3}{12} = \dfrac{11}{12}.

Worked example: 56−38\dfrac{5}{6} - \dfrac{3}{8}

  1. lcm(6,8)=24\text{lcm}(6, 8) = 24.
  2. 56=2024\dfrac{5}{6} = \dfrac{20}{24} and 38=924\dfrac{3}{8} = \dfrac{9}{24}.
  3. Difference: 20−924=1124\dfrac{20 - 9}{24} = \dfrac{11}{24}.

Multiplying

ab⋅cd=a⋅cb⋅d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

No common denominator needed. Canceling before multiplying (cross-canceling) avoids large numbers.

Worked example: 34⋅89\dfrac{3}{4} \cdot \dfrac{8}{9}

  1. Cross-cancel: 33 with 99 (divide by 3) and 88 with 44 (divide by 4).
  2. You get 11⋅23\dfrac{1}{1} \cdot \dfrac{2}{3}.
  3. Result: 23\dfrac{2}{3}.

Without canceling first: 2436\dfrac{24}{36}, which also reduces to 23\dfrac{2}{3}, with more work.

Dividing

Multiply the first fraction by the reciprocal of the second:

ab÷cd=ab⋅dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Worked example: 56÷109\dfrac{5}{6} \div \dfrac{10}{9}

  1. Flip the second fraction: 56⋅910\dfrac{5}{6} \cdot \dfrac{9}{10}.
  2. Cross-cancel: 55 with 1010 (by 5) and 99 with 66 (by 3).
  3. 12⋅32=34\dfrac{1}{2} \cdot \dfrac{3}{2} = \dfrac{3}{4}.

Mixed numbers

Convert to an improper fraction before operating: 213=2⋅3+13=732\tfrac{1}{3} = \dfrac{2 \cdot 3 + 1}{3} = \dfrac{7}{3}.

Worked example: 213+1122\tfrac{1}{3} + 1\tfrac{1}{2}

  1. 73+32\dfrac{7}{3} + \dfrac{3}{2}.
  2. lcm(3,2)=6\text{lcm}(3, 2) = 6: 146+96=236\dfrac{14}{6} + \dfrac{9}{6} = \dfrac{23}{6}.
  3. Back to a mixed number: 23=3⋅6+523 = 3 \cdot 6 + 5, so 3563\tfrac{5}{6}.

Common mistakes

  1. Adding the denominators.
  2. Flipping the first fraction when dividing instead of the second.
  3. Canceling terms of a sum: in 2+62\dfrac{2 + 6}{2} you cannot “cancel” the 2; the value is 44.

Frequently asked questions

Can I add fractions by adding numerators and denominators?

No. 1/2 + 1/2 would give 2/4 = 1/2, when the correct answer is 1. You only add numerators after making the denominators equal.

Do I need the least common denominator, or does any common denominator work?

Any common multiple works, including the product of the denominators. The least one just keeps numbers smaller and reduces simplification at the end.

Why does dividing by a fraction mean multiplying by its reciprocal?

Dividing by b/c asks how many times b/c fits into the number, and multiplying by c/b gives exactly that, since (b/c) · (c/b) = 1.