Adding and subtracting need a common denominator
Fractions only add or subtract directly when they already share a denominator. You cannot combine "3 of something cut into 4 pieces" with "1 of something cut into 6 pieces" until both are re-cut into the same size piece. The fix is to rewrite each fraction as an equivalent fraction over a shared denominator (multiplying a fraction's numerator and denominator by the same number never changes its value), then add or subtract the numerators. The Common Core State Standards for Mathematics describe exactly this in the Grade 5 fractions domain, standard 5.NF.A.1: "replacing given fractions with equivalent fractions… to produce an equivalent sum or difference of fractions with like denominators." Their own worked example is 2/3 + 5/4 = 8/12 + 15/12 = 23/12, found by cross-multiplying the two denominators (3×4=12) instead of hunting for the least common multiple by hand.
A subtraction example: 3/4 − 1/6. The shared denominator is 4×6=24, so 3/4 becomes 18/24 and 1/6 becomes 4/24, leaving 18/24 − 4/24 = 14/24, which simplifies to 7/12 (dividing both terms by their greatest common divisor, 2).
Multiplying and dividing skip the common denominator entirely
Multiplication of fractions is a straight-across operation: multiply the two numerators together, multiply the two denominators together, and simplify. 2/3 × 3/4 = (2×3)/(3×4) = 6/12, which reduces to 1/2. No common denominator is needed, because multiplication is not measuring "how many of the same-size piece" the way addition is. It is scaling one fraction by another.
Division works by flipping the second fraction (its reciprocal) and multiplying: 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. That matches the intuitive version too: "how many 1/4-cup scoops fit in 1/2 cup" is 2. Multiplying by a fraction's reciprocal is how division is formally defined for fractions in the Common Core standards (6.NS.A.1 extends this to dividing any fraction by any fraction). Dividing by a fraction that equals zero, like 0/5, has no defined answer, the same way dividing by the whole number 0 does not.
Simplifying to lowest terms
A fraction is in "lowest terms" when its numerator and denominator share no common factor greater than 1. Finding that shared factor, the greatest common divisor (GCD), is done here with the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is 0. The last non-zero remainder is the GCD. This method is over two thousand years old. Euclid's Elements (Book VII, Proposition 2) describes it as a way to find the "greatest common measure" of two numbers, and it is still the standard algorithm computers use for this today. For 36/48, the GCD of 36 and 48 is 12, so dividing both terms by 12 gives 3/4.
A fraction where the numerator is 0, like 0/9, always simplifies to 0/1. Zero pieces of anything is zero, regardless of the denominator.
Mixed numbers are fractions in disguise
A mixed number like 2 3/4 means "2 whole units, plus 3/4 of another unit," which is the same quantity as the improper fraction 11/4 (an improper fraction is simply one whose numerator is larger than its denominator). To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator: (2×4)+3=11, over 4. To go the other way, divide the numerator by the denominator. The whole-number quotient becomes the whole part, and the remainder becomes the new numerator over the same denominator: 11÷4 is 2 remainder 3, so 11/4 becomes 2 3/4.
A recipe example: a batch of bread dough calls for 2 3/4 cups of flour and you want to double it. Convert to an improper fraction first (2 3/4 = 11/4), then multiply by 2/1: 11/4 × 2/1 = 22/4, which simplifies to 11/2, or 5 1/2 cups. Trying to double "2 3/4" directly, without converting to an improper fraction first, is where most doubled-recipe arithmetic mistakes happen.