How to work with fractions

Fractions feel harder than they are because the rules change depending on whether you’re adding, multiplying, or converting to a decimal. Adding 1/3 and 1/4 requires a common denominator; multiplying them doesn’t. Simplifying 24/36 requires finding a GCD; converting 1/7 to a decimal gives you an infinitely repeating pattern that confuses people into thinking they made an arithmetic error. Mixed numbers like 2 3/4 hide an extra step where you either convert to an improper fraction or keep them separate. This guide covers the vocabulary (proper, improper, mixed), the four operations with worked examples, simplification using the greatest common divisor, decimal conversion including repeating decimals, and the tricks that make fraction arithmetic feel mechanical instead of mysterious.

Vocabulary

A fraction has a numerator on top and a denominator on bottom. Proper fractions have a numerator smaller than the denominator (3/4, 2/7). Improper fractions have numerator larger or equal (7/4, 5/5).Mixed numbers combine a whole number and a proper fraction (1 3/4). Every improper fraction can be rewritten as a mixed number and vice versa.

7/4 = 1 3/4         (improper → mixed: divide 7÷4 = 1 remainder 3)
2 1/3 = 7/3         (mixed → improper: 2×3 + 1 = 7)
5/5 = 1             (improper that equals a whole number)

Simplifying with GCD

Divide numerator and denominator by their greatest common divisor. For 24/36, GCD = 12, so 24/36 = 2/3. For 15/25, GCD = 5, so 15/25 = 3/5. The Euclidean algorithm finds GCD quickly: repeatedly replace the larger number with the remainder of the division. Always simplify at the end unless the problem asks for a specific form.

24/36  → gcd 12 → 2/3
15/25  → gcd 5  → 3/5
100/75 → gcd 25 → 4/3 → 1 1/3
42/56  → gcd 14 → 3/4

Adding and subtracting: you need a common denominator

You can’t add 1/3 and 1/4 directly. Find the least common denominator (LCD), rewrite both fractions over that denominator, then add numerators. For 1/3 and 1/4, LCD = 12, so 1/3 = 4/12 and 1/4 = 3/12, giving 7/12.

1/3 + 1/4
  LCD = 12
  = 4/12 + 3/12
  = 7/12

5/6 − 1/4
  LCD = 12
  = 10/12 − 3/12
  = 7/12

When the denominators share no common factors, the LCD is their product. When they share factors, find the LCD by multiplying each denominator’s unique prime factors at their highest power.

Multiplying: just multiply across

Multiply numerators together and denominators together, then simplify: a/b × c/d = (ac)/(bd). No common denominator needed. For mixed numbers, convert to improper first.

2/3 × 4/5 = 8/15
1/2 × 3/4 = 3/8
1 1/2 × 2/3 = 3/2 × 2/3 = 6/6 = 1

Shortcut: cancel before multiplying
  4/9 × 3/8 = (4/8) × (3/9) = 1/2 × 1/3 = 1/6

Dividing: multiply by the reciprocal

Flip the second fraction and multiply: a/b ÷ c/d = a/b × d/c = ad/bc. This is the rule that feels arbitrary but falls out of the definition of division as multiplying by the inverse.

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2
3/4 ÷ 6    = 3/4 × 1/6 = 3/24 = 1/8

Converting to decimals

Divide numerator by denominator. Some fractions terminate (3/4 = 0.75); others repeat (1/3 = 0.3333..., 1/7 = 0.142857142857...). A fraction terminates if and only if its simplified denominator’s prime factors are only 2 and 5. Everything else repeats.

1/2  = 0.5          (terminates; denom = 2)
3/8  = 0.375        (terminates; denom = 8 = 2³)
1/3  = 0.333...     (repeats; denom = 3)
1/7  = 0.142857...  (6-digit repeat block)
1/11 = 0.090909...
1/6  = 0.1666...

Converting a repeating decimal back to a fraction

Let x = 0.3333.... Then 10x = 3.3333..., and subtracting gives 9x = 3, so x = 3/9 = 1/3. For a longer repeat block, multiply by 10^n where n is the repeat length: for 0.142857142857..., multiply by 10^6 and subtract.

Comparing fractions

Three techniques. First, convert to a common denominator and compare numerators. Second, cross-multiply: a/b versus c/d—whichever of adand bc is larger tells you the answer. Third, convert to decimals (fastest for a pile of fractions, slower for exact comparison).

Is 5/8 > 3/5?
  Cross-multiply: 5×5 = 25 vs 3×8 = 24
  25 > 24, so 5/8 > 3/5  ✓

Mixed numbers: two approaches

For addition, you can keep the whole parts separate: 2 1/3 + 1 1/4 = 3 + (1/3 + 1/4) = 3 + 7/12 = 3 7/12. For multiplication and division, always convert to improper first—the “FOIL” approach (multiplying mixed numbers directly) is a classic trap.

Common mistakes

Adding numerators and denominators directly. 1/3 + 1/4 is NOT 2/7. You must find a common denominator first. This is the single most common error.

Forgetting to flip when dividing. 2/3 ÷ 4/5 is NOT 8/15; that’s the product. Division requires you to flip the second fraction, then multiply.

Multiplying mixed numbers without converting. 2 1/2 × 3 1/3 is NOT 6 1/6. Convert each to an improper fraction first: 5/2 × 10/3 = 50/6 = 25/3 = 8 1/3.

Over-simplifying before multiplying. You can cancel a factor from any numerator against any denominator across a multiplication, but not across addition or subtraction. Don’t cancel diagonally in 2/3 + 3/4.

Confusing a repeating decimal with a rounding error. 1/3 really is 0.3333... with 3s forever, not 0.3333333 as a finite digit string. When a calculator shows 0.33333333, it’s truncating.

Writing 0 as a denominator. Division by zero is undefined. 3/0is not a number; any chain of operations that leads to /0 means you took a wrong turn.

Leaving answers unsimplified. Most teachers and most real-world contexts expect simplified form. 6/8 should be 3/4; 10/4 should be 5/2 or 2 1/2.

Run the numbers

Skip the common-denominator arithmetic and let our fraction calculator handle the steps with show-your-work output. Pair it with the ratio calculator when the problem is really a proportion in disguise, and the percentage calculator for the common case of converting a fraction into a percent.