Quick answer

LCD(b, d) = lcm(b, d). For nonzero integers, lcm(b, d) = |b × d| / gcd(b, d). Scale numerators after the LCD is found.

Formula

  • LCD = lcm(b, d)
  • lcm = |b × d| / gcd(b, d)
  • n/d → (n × LCD/d) / LCD

Introduction

Every LCD problem begins with denominators. Numerators matter only after denominators match.

The formula ties LCD directly to lcm. That single relationship drives equivalent fractions, addition, and subtraction.

You can compute lcm by listing multiples, using prime factorization, or applying the gcd shortcut. Each route should give the same LCD when done correctly.

If the vocabulary is new, review what is the least common denominator before applying the formulas below.

Formula explanation

Start with two denominators b and d. The LCD is the smallest number divisible by both.

Listing multiples works when denominators are small. For 8 and 12, multiples of 12 are 12, 24, 36. Since 8 divides 12, lcm = 12.

Prime factorization helps when factors are messy. Factor 8 = 2³ and 12 = 2² × 3. Take the highest powers: 2³ and 3¹. LCD = 8 × 3 = 24.

The gcd shortcut is fast on tests: lcm(8, 12) = 96 / gcd(8, 12) = 96 / 4 = 24. Same result as the prime grid method.

When denominators share many prime factors, the dedicated walkthrough in finding LCD with prime factorization shows a full factor tree from start to finish.

Methods compared

  • Multiples: list until a shared multiple appears
  • Primes: multiply highest prime powers
  • gcd shortcut: |b × d| / gcd(b, d)

After LCD is known, scale each fraction. For a/b, multiply numerator and denominator by LCD/b so the new denominator equals LCD.

Example scaling: 3/8 with LCD 24 becomes (3×3)/(8×3) = 9/24. The fraction still represents the same amount on a number line.

Choosing the LCD instead of the product bd avoids oversized numerators. For 6 and 8, bd = 48 but lcm is 24, which is easier to manage in later addition steps.

Step-by-step breakdown

  1. Write denominators only Ignore numerators at first. For mixed numbers, read the denominator of the fractional part.
  2. Compute lcm using one method Pick multiples, primes, or gcd based on the numbers you see.
  3. Label the lcm as LCD State LCD = lcm(b, d) clearly on homework so the teacher sees that step.
  4. Build equivalent fractions Multiply each numerator by LCD divided by its original denominator.
  5. Proceed with the operation Add or subtract numerators, or compare numerators, using the LCD as the shared bottom.

Example: 5/6 and 1/4

gcd(6, 4) = 2. lcm = (6×4)/2 = 12. LCD = 12.

Scale: 5/6 = 10/12 and 1/4 = 3/12. If adding, 10/12 + 3/12 = 13/12 = 1 1/12.

Confirm the lcm line and equivalents at /#calculator.