Quick answer

Find LCD, rewrite fractions at LCD, add numerators, keep LCD, then simplify the sum if possible.

Formula

  • LCD = lcm(b, d)
  • a/b + c/d → (scaled a + scaled c) / LCD
  • Simplify with gcd at the end

Introduction

Addition with fractions only works when both fractions refer to the same-sized parts. The LCD is the preferred common denominator because it keeps numerators as small as possible.

A common mistake is adding denominators. After you find the LCD, only the top numbers combine.

Another mistake is skipping the equivalent fraction step. Writing 1/2 + 1/3 as 2/5 ignores the meaning of denominators entirely.

Before adding, make sure you can find the LCD quickly. The walkthroughs in least common denominator examples give you fraction pairs to practice.

Common denominator conversion

Scale each fraction: new numerator = old numerator × (LCD ÷ old denominator). The denominator becomes the LCD.

Example: 1/5 + 1/2. LCD = 10. Work: 2/10 + 5/10 = 7/10.

Mixed numbers: convert to improper form if needed, find LCD on fractional denominators, scale, then add.

When the sum is improper, convert to a mixed number for a clearer answer, such as 13/8 = 1 5/8.

The symbolic template for the LCD step is written out in the least common denominator formula article if you want the rule in one place.

Simplification after adding

  • Sum numerators at LCD
  • Check gcd(numerator, denominator)
  • Write improper or mixed form

After you add numerators, the result may not be in lowest terms. Divide top and bottom by their gcd.

Example: 2/8 + 4/8 = 6/8 = 3/4 after simplifying by 2.

On homework, show the unsimplified sum first, then the simplified answer. That shows your teacher both steps.

Addition steps

  1. Find the LCD Use multiples, primes, or gcd on the denominators only.
  2. Write equivalent fractions Both fractions must show the LCD as the denominator.
  3. Add numerators Keep the LCD. Write the raw sum before simplifying.
  4. Simplify the result Apply gcd reduction. Convert to mixed form if required.
  5. Check with the calculator Confirm LCD and equivalents before you add on paper.

Example: 2/3 + 1/6

Denominators 3 and 6. Since 6 is a multiple of 3, LCD = 6.

Scale: 2/3 = 4/6. Sum: 4/6 + 1/6 = 5/6. Already in lowest terms.

Enter 2, 3, 1, 6 in the /#calculator to verify equivalents, then add on paper.