Adding Fractions Calculator
Result
Sum (simplest form)
- Common denominator
- 12
- First fraction, converted
- 3/12
- Second fraction, converted
- 2/12
An adding fractions calculator adds two fractions and shows the step that makes it possible: the two fractions are rewritten over a common denominator first, and only then are the numerators added. Adding 1/4 and 1/6 gives 5/12, and the working is 3/12 plus 2/12: a quarter is three twelfths and a sixth is two twelfths, so the total is five twelfths. That rewriting step is the whole subject. Fractions can be added directly when their denominators already match, because the denominator is naming the size of the pieces and the numerators are counting them: 1/5 plus 2/5 is three fifths, the same way one apple plus two apples is three apples. Unlike denominators name pieces of different sizes, and counting them together would be adding apples to oranges, so each fraction is first converted into an equal fraction with the same denominator. The common denominator used here is the least common multiple of the two denominators, which is the smallest number both divide into: for 4 and 6 that is 12 and not 24. Multiplying the denominators together always gives a valid common denominator, and it is what a four-function fraction calculator uses, but the pieces it produces are not the smallest ones and the extra factor has to be divided out again afterwards. The smallest common denominator also keeps the converted numerators as small as they can be, though it does not guarantee that the sum is already in lowest terms: 1/3 plus 1/6 becomes 2/6 plus 1/6, which is 3/6 and then 1/2, and 1/2 plus 1/2 becomes 2/2 and then 1. The sum is simplified before it is printed, and the two converted fractions beside it are left as they are, because they are the step being shown. The result is printed as a fraction, improper ones included: adding 3/4 and 5/8 gives 11/8 rather than a mixed number, so that every figure on the panel is written the same way. Both numerators and denominators are whole numbers here, and both denominators have an upper limit, because the arithmetic is done exactly in integers and a denominator in the hundreds of thousands is already far past anything a page needs to add up.
Common additions and their common denominators
| First fraction | Second fraction | Common denominator | Sum |
|---|---|---|---|
| 1/4 | 1/6 | 12 | 5/12 |
| 1/2 | 1/3 | 6 | 5/6 |
| 1/4 | 1/2 | 4 | 3/4 |
| 1/3 | 1/6 | 6 | 1/2 |
| 2/3 | 1/6 | 6 | 5/6 |
| 3/4 | 5/8 | 8 | 11/8 |
| 1/7 | 1/11 | 77 | 18/77 |
| 1/2 | 1/2 | 2 | 1 |
The first row is the default case and the clearest example of what the least common multiple buys: 1/4 and 1/6 give a common denominator of 12, not 24. The second row is the case where the two denominators share no factor, so the common denominator really is the product, and it is the row that makes the shortcut look universal. The third row has one denominator dividing the other, so the common denominator is simply the larger one. The fourth row is the one that shows the sum still needing work even with the least common denominator: 1/3 plus 1/6 is 2/6 plus 1/6, and 3/6 has to be reduced to 1/2. The second and fifth rows both come to 5/6 by different routes, which is worth noticing: 1/2 plus 1/3 and 2/3 plus 1/6 both convert to sixths and both add to five of them, so the same answer arrives from two unrelated pairs of fractions. The sixth row is an improper result left improper, the seventh is a pair with no common factor at all, giving a common denominator of 77, and the last row is two halves making a whole, which the panel prints as 1. Every row comes out of the same formula, and no row is a special case of it.
Formula
Common denominator = least common multiple of the two denominators Converted = each numerator × (common denominator ÷ its own denominator) Sum = (converted first + converted second) ÷ common denominator, simplified
- First fraction
- The first numerator and denominator. The denominator may not be zero, and the numerator is the one that carries a minus sign if the fraction is negative.
- Second fraction
- The second numerator and denominator, treated exactly like the first: the order of the two addends does not change the sum.
- Common denominator
- The least common multiple of the two denominators: 12 for 1/4 plus 1/6. It is the smallest number that both denominators divide into, and it is what makes the numerators countable together.
- Converted fractions
- Each fraction rewritten over the common denominator, which means multiplying its numerator by the same factor its denominator was multiplied by: 1/4 becomes 3/12 and 1/6 becomes 2/12. The values are unchanged; only the size of the pieces is.
- Sum
- The two converted numerators added together over the common denominator, then reduced to simplest form. It is printed as a fraction, and an improper one is left improper rather than being turned into a mixed number.
Two fractions are added with this method whenever the denominators differ, which is most of the time in practice. The first use is school work, where the point is the conversion step rather than the answer: the panel prints the two converted fractions so that the working can be checked against a page of homework instead of only the final figure. The second is a measurement that comes in fractions of different sizes, such as a quarter of an hour plus a sixth of an hour, where the sum only means something once both are counted in the same unit, in this case twelfths of an hour. The third is checking someone else's arithmetic: a total that came out as 10/24 for a quarter plus a sixth was added by multiplying the denominators, which is correct but unreduced, and this page's common denominator of 12 shows that the same sum can be done with smaller pieces. When the denominators already match, the conversion step is a no-op and the numerators are simply added.
Worked examples
1/4 + 1/6
- Least common multiple of 4 and 6: 12
- Convert the first: 1/4 = 3/12, because 12 ÷ 4 = 3
- Convert the second: 1/6 = 2/12, because 12 ÷ 6 = 2
- Add the numerators: 3 + 2 = 5, giving 5/12
The default case, and the one that shows what the common denominator buys: the pieces are twelfths, not twenty-fourths, because 12 is the smallest number both 4 and 6 divide into. The sum is already in lowest terms.
1/2 + 1/3
- Least common multiple of 2 and 3: 6
- Convert the first: 1/2 = 3/6
- Convert the second: 1/3 = 2/6
- Add the numerators: 3 + 2 = 5, giving 5/6
The common denominator here happens to be the product of the two denominators, because 2 and 3 share no factor. That is the one case where multiplying the denominators gives the same answer as the least common multiple, and it is why the shortcut looks like it always works.
3/4 + 5/8, an improper result
- Least common multiple of 4 and 8: 8, since 8 is already a multiple of 4
- Convert the first: 3/4 = 6/8
- Convert the second: 5/8 is already in eighths
- Add the numerators: 6 + 5 = 11, giving 11/8
One denominator divides the other, so the common denominator is the larger of the two rather than the product. The total is more than one, and it is printed as 11/8: an improper fraction is a perfectly good way to write that, and it keeps every figure in the same notation.
1/2 + 1/2, which comes out whole
- The denominators already match, so the common denominator is 2
- Convert: both fractions are already in halves
- Add the numerators: 1 + 1 = 2, giving 2/2
- Simplify: 2/2 is 1
Two halves make a whole, and the panel writes it as 1 rather than 2/2 or 1/1. The simplification step is not optional even when the common denominator is the least one: the converted fractions are in lowest terms, but the sum is not automatically.
Limitations
Both inputs must be whole numbers, so 0.5 plus 1/3 is not something this page will add: convert the decimal to a fraction first, and the reason is that the whole calculation is done in exact integer arithmetic, which stops being exact once a decimal is involved. Denominators must be positive and are limited, and numerators are limited too, because the two converted numerators are multiplied up before they are added. The result is always written as a fraction and never as a mixed number, so a sum of 11/8 stays 11/8 rather than becoming 1 3/8. The common denominator is the least common multiple, which is a different intermediate step from the one a four-function fraction calculator shows: adding the same pair there gives 6/24 and 4/24 rather than 3/12 and 2/12, and both are correct, they just make the pieces differently sized. Only addition is done here: subtraction, multiplication and division are separate pages, and a negative result is written with the minus sign on the numerator. Finally, the page adds fractions and nothing else, so a mixed number has to be turned into an improper fraction before it can be typed in.
Frequently asked questions
- How do I add fractions with different denominators?
- Rewrite both fractions so that they share a denominator, then add the numerators and keep that denominator. That is the whole formula: for 1/4 and 1/6 the shared denominator is 12, so they become 3/12 and 2/12 and the total is 5/12.
- How is the common denominator worked out?
- It is the least common multiple of the two denominators: the smallest number both of them divide into. For 4 and 6 that is 12. Multiplying the two denominators together always gives a workable answer as well, but it is often larger than it needs to be.
- Why not just multiply the denominators together?
- You can, and the answer will be right, but the pieces will be smaller than necessary: 1/4 plus 1/6 done that way gives 6/24 plus 4/24, which is 10/24, and that still has to be reduced to 5/12. Using the least common multiple does the same job with twelfths instead of twenty-fourths.
- Does the answer need simplifying afterwards?
- Sometimes, and the page does it before printing the result. Using the least common denominator keeps the pieces as large as possible, but it does not promise that the numerators will add up to something with no factor in common: 1/3 plus 1/6 is 2/6 plus 1/6, which is 3/6 and simplifies to 1/2.
- Can the result be an improper fraction?
- Yes, and it is printed that way: 3/4 plus 5/8 is 11/8, which is more than one. Writing it as a mixed number is a separate conversion, and this page keeps every figure in the same numerator-over-denominator form.
- What are the two converted fractions on the panel?
- They are the two addends rewritten over the common denominator, which is the step the addition depends on: 1/4 becomes 3/12 and 1/6 becomes 2/12. Their values have not changed, only the size of the pieces they are counted in.
- Can I add a decimal and a fraction here?
- No. Both numerators and both denominators have to be whole numbers, so 0.5 plus 1/3 is rejected. The reason is that the conversion and the addition are done in exact integer arithmetic, and converting the decimal to a fraction first keeps it that way.
References
- Fraction — the common denominator of two fractions is a common multiple of their denominators, and the least common denominator is the least of those; the entry works an addition through the conversion step that this page prints — Wolfram MathWorld (United States)
- Least Common Multiple — the smallest positive integer that both numbers divide, which is the common denominator this page uses, computed from the greatest common divisor rather than by listing multiples — Wolfram MathWorld (United States)
- Greatest Common Divisor — the largest integer dividing both numbers, which is what the least common multiple is derived from and what the sum is reduced by at the end — Wolfram MathWorld (United States)
- Reduced Fraction — a fraction written with the common factors of numerator and denominator cancelled; the converted fractions on this page are already in that form, while the sum may still need it — Wolfram MathWorld (United States)