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Fraction addition

How to Add 3 Fractions

A third addend looks intimidating, but the method does not grow with the problem: the common denominator must suit all three fractions, every numerator joins over it, and the finishing touches are the same simplify-and-regroup pair as always. The tool below takes three terms live.

  • LCM of all three
  • add the numerators in one pass
  • order does not matter
  • same rule, more terms
Start with the basics of adding fractions

Visual addition tool

See how fraction parts become ready to add

Start with one problem, make the parts match when needed, then combine them.

Type the full expression

Examples: 1/4 + 2/3, 2 + 3/4, 1 1/2 + 2/3, or 1/6 + 2/6 + 3/6.

Visual explanation

Notice the part sizes

notice

The fractions name pieces of different sizes, so they cannot be combined yet.

First addend1/2
Second addend1/3
Third addend1/4
1/2 part
1/3 part

These pieces are not the same size yet.

Quick answer

One method, three terms

  1. 1. Find one common denominator for all three. The least common multiple of every denominator: for 1/2 + 1/3 + 1/4 it is 12.
  2. 2. Rewrite each fraction, then add in one pass. 1/2 + 1/3 + 1/4 becomes 6/12 + 4/12 + 3/12, and the numerators sum to 13.
  3. 3. Simplify and regroup. 13/12 shares no factor, but it tops one whole: the answer is 1 1/12.

The method: nothing new, just more terms

Adding three fractions is the two-fraction method with one extra guest at the table. The common denominator now has to be a multiple of all three denominators — the least common multiple of 2, 3, and 4 is 12 — and every fraction is renamed onto it before any adding happens:

1/2 + 1/3 + 1/4

= 6/12 + 4/12 + 3/12

= 13/12

= 1 1/12

Watch where the multipliers come from: 12 ÷ 2 = 6 stretches 1/2 to 6/12, 12 ÷ 3 = 4 stretches 1/3 to 4/12, and 12 ÷ 4 = 3 stretches 1/4 to 3/12. Each fraction follows the same stretch rule it always did — there is just one more of them.

The tool above runs the whole chain live for three terms: try the preset 1/2 + 1/3 + 1/4 to see all three rewrite arrows, then switch to 1/6 + 2/6 + 3/6 and watch the rewrite step vanish.

Like denominators: three terms in one line

When all three denominators already match, the problem is a single counting sentence. Three sixths of a bar plus two sixths plus one sixth:

1/6 + 2/6 + 3/6

= 6/6

= 1

One sixth, two sixths, three sixths — six sixths in total, and six sixths is exactly one whole. No rewrite, no common denominator hunt: the only work is the sum 1 + 2 + 3 and the final regroup.

This is the version youngest students meet first, and it is worth saying out loud: “one sixth, add two more sixths, add three more sixths — how many sixths now?” The counting sentence is the entire method.

Order and grouping do not matter

Two properties of addition make three-term problems flexible. Commutativity: the terms can sit in any order — 1/2 + 1/3 + 1/4 equals 1/4 + 1/3 + 1/2. Associativity: you may add any two first and then the third — (1/2 + 1/3) + 1/4 and 1/2 + (1/3 + 1/4) land on the same 13/12.

That is why “do I add all three at once, or two first?” has no wrong answer. Adding all three numerators in one pass is faster; pairing two friendly fractions first is a legitimate strategy when the rewrite feels crowded. What is never optional is the common denominator covering all three terms before any numerator moves.

A practical ordering tip: scanning the three denominators for a multiple relationship first — as in 2/5 + 1/10 + 3/10, where 10 already covers 5 — often shrinks the rewrite to a single fraction.

Common mistakes when adding 3 fractions

  • Finding the LCM of only two of the three denominators: With 1/2 + 1/3 + 1/4, taking the LCM of 2 and 3 alone gives 6 — but 4 does not divide 6, so fourths can never be renamed onto it. Every denominator in the problem must divide the common denominator. For 2, 3, 4 the answer is 12.
  • Renaming only two of the three fractions: After finding the common denominator, every single term must be restated over it — 1/2 + 1/3 + 1/4 becomes 6/12 + 4/12 + 3/12, all three. Combining a rewritten fraction with an un-rewritten one mixes piece sizes and produces a wrong count, no matter how carefully the addition is done.
  • Dropping a numerator on the way: Three addends give three chances to lose track. After the rewrite, the numerators all add in one pass — 6 + 4 + 3 = 13 — whether you sum them left to right or group them first. Write all three rewritten numerators down before adding so nothing disappears.

Worked example

Three matching denominators: 1/6 + 2/6 + 3/6

1/6 + 2/6 + 3/6 = 6/6 = 1

All three fractions already count sixths, so the numerators add in one pass: 1 + 2 + 3 = 6. Six sixths is exactly one whole, and the regrouping step turns 6/6 into 1.

Worked example

Three different denominators: 1/2 + 1/3 + 1/4

1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12 = 1 1/12

The least common multiple of 2, 3, and 4 is 12 — every denominator must divide it. Each fraction stretches: 1/2 becomes 6/12, 1/3 becomes 4/12, 1/4 becomes 3/12. The numerators sum to 13, one more than a whole, so the answer is 1 1/12.

Worked example

One denominator already fits: 2/5 + 1/10 + 3/10

2/5 + 1/10 + 3/10 = 4/10 + 1/10 + 3/10 = 8/10 = 4/5

When one denominator is a multiple of the others, the largest denominator is already the common one. Only 2/5 needs rewriting — it doubles to 4/10 — and the three numerators add to 8, which simplifies with 10 down to 4/5.

Worked example

A mixed number joins: 1 1/2 + 1/3 + 1/6

1 1/2 + 1/3 + 1/6 = 9/6 + 2/6 + 1/6 = 12/6 = 2

Rewrite the mixed number as the improper 3/2 first — in sixths that is 9/6. The common denominator is 6, the numerators sum to 9 + 2 + 1 = 12, and 12/6 is exactly 2 whole things.

Keep learning

Related fraction topics

Return to Adding FractionsAdding and Subtracting with Unlike DenominatorsAdding Fractions Examples

Common questions

Adding 3 Fractions FAQ

The same way as two fractions, with one change: the least common denominator must work for all three denominators. For 1/2 + 1/3 + 1/4 it is 12. Rewrite each fraction over 12 (6/12, 4/12, 3/12), add all three numerators (13), keep the denominator, then simplify and regroup: 1 1/12.