Method · Fractions
How to Add Fractions With Different Denominators
Adding fractions is only hard when the denominators differ. Once you know why they have to match, the method stops being a rule to memorise.
Why you cannot just add across
The first thing to understand about how to add fractions is why 1/2 + 1/3 is not 2/5. A fraction is a count of parts of a particular size. The denominator says how big each part is; the numerator says how many you have. Halves and thirds are different sizes, so counting them together means nothing until they are made the same size.
Adding across the bottom does something worse than give a wrong answer — it gives an answer smaller than one of the fractions you started with. 2/5 is less than 1/2, which is impossible when you have added something positive to it. That sanity check catches the error every time.
Adding fractions with the same denominator
When the parts are already the same size, add the numerators and leave the denominator alone. Three eighths plus two eighths is five eighths, in the same way that three apples plus two apples is five apples.
- 3⁄8 + 2⁄8the parts are already the same size
- 5⁄8add the numerators, keep the denominator
The denominator does not change because the size of each part has not changed. Only the count has.
Adding fractions with unlike denominators
When the denominators differ, rewrite both fractions over a shared denominator first. Any common multiple works, but the least common denominator keeps the numbers small and usually saves you a simplification at the end.
- 2⁄3 + 1⁄4thirds and quarters are different sizes
- 12the least common denominator — the smallest multiple of both 3 and 4
- 8⁄12 + 3⁄12multiply each fraction top and bottom to reach twelfths
- 11⁄12now the parts match, so add the numerators
Notice what happened to each fraction: 2/3 was multiplied top and bottom by 4, and 1/4 by 3. Multiplying both parts of a fraction by the same number produces an equivalent fraction — the value is untouched, only the way it is written has changed.
Finding the least common denominator
The least common denominator of two fractions is the least common multiple of their denominators. For small numbers, list the multiples until one appears in both lists:
- Multiples of 3: 3, 6, 9, 12, 15
- Multiples of 4: 4, 8, 12, 16
- The first shared value is 12, so 12 is the least common denominator.
For larger denominators, multiply them together and then simplify at the end. It is never wrong — 3 × 4 = 12 here anyway — it just occasionally leaves bigger numbers to reduce.
Adding mixed numbers
A mixed number like 2¾ is a whole number and a fraction stuck together. There are two workable approaches, and the second is more reliable under pressure.
- Add the whole parts and the fractional parts separately, then recombine — quick, but you have to remember to carry when the fractions sum past 1.
- Convert both to improper fractions first, add normally, then convert back — more writing, but no carrying step to forget.
To convert 2¾ to an improper fraction, multiply the whole number by the denominator and add the numerator: 2 × 4 + 3 = 11, giving 11/4.
Always simplify the result
An answer of 8/12 is correct but unfinished. Divide the numerator and denominator by their greatest common factor — here 4 — to reach 2/3, the same number in lowest terms.
If the result is top-heavy, like 11/4, convert it back to a mixed number when the question was asked in mixed numbers. Match the form of the question unless told otherwise.
Questions about how to add fractions
Why do you need a common denominator to add fractions but not to multiply them?
Adding combines parts, so the parts must be the same size first — that is what the common denominator creates. Multiplying takes a fraction of a fraction rather than combining like parts, so no matching is required.
What is the quickest way to find a common denominator?
Multiply the two denominators together. It always produces a valid common denominator and needs no searching. The only cost is that the numbers may be larger than necessary, so you may have to simplify at the end.
How do you add three or more fractions?
Exactly the same way. Find one denominator that all of them divide into, rewrite every fraction over it, then add all the numerators in one go.
Do I have to simplify my answer?
Unless the question says otherwise, yes. 8/12 and 2/3 are the same number, but marked work almost always expects the lowest-terms form.