The short answer
You can only add fractions that are cut into the same size of pieces. So rewrite both fractions over a shared denominator, add the numerators and leave the denominator alone, then divide the top and bottom by their greatest common divisor.
Written as one formula: a/b + c/d = (a·d + c·b) / (b·d), reduced afterwards.
The product b·d always works as a common denominator, which is why that formula never fails. Using the lowest common multiple instead just saves you some reducing at the end — it never changes the answer.
The five steps
- Check the denominators. If they already match, skip to step 4.
- Find a common denominator. Multiplying them together is always safe. The lowest common multiple is tidier: for 8 and 12 it is 24, not 96.
- Rescale each fraction. Multiply the numerator and the denominator by the same number, so the value does not change. 3/8 becomes 9/24 because 24 ÷ 8 = 3, and 3 × 3 = 9.
- Add the numerators. The denominator stays as it is. It names the size of the pieces, and the pieces have not changed size.
- Reduce, then convert if you want to. Divide top and bottom by their greatest common divisor. If the numerator is larger than the denominator, you can rewrite the result as a whole number plus a fraction.
Subtraction is identical with a minus sign at step 4. Multiplication needs no common denominator at all — multiply across the top and across the bottom — and division means multiplying by the reciprocal of the second fraction.
Worked examples
Different denominators: 1/2 + 1/3
The lowest common multiple of 2 and 3 is 6. Rescale: 1/2 = 3/6 and 1/3 = 2/6. Add the numerators: 3 + 2 = 5, so the answer is 5/6, which is 0.833333 to six places. Nothing reduces, because 5 and 6 share no factor above 1.
Notice what did not happen: the denominators were not added. 1/2 + 1/3 is not 2/5. A quick sanity check catches this — 2/5 is 0.4, which is smaller than the 1/2 you started with, so it cannot be right.
Bigger denominators: 3/8 + 5/12
Eight and twelve both divide into 24, so 24 is the lowest common multiple. Rescale by 3 and by 2: 3/8 = 9/24 and 5/12 = 10/24. Adding gives 19/24, which is already reduced. The answer is 19/24, or 0.791667.
Multiplying the denominators instead gives 96 as the common denominator: 36/96 + 40/96 = 76/96. That reduces, by 4, to the same 19/24. Both routes are correct; the second one just leaves more tidying up.
Mixed numbers: 2¾ + 1⅚
Convert each mixed number to a single fraction before you do anything else. 2¾ is (2 × 4 + 3)/4 = 11/4, and 1⅚ is (1 × 6 + 5)/6 = 11/6. The lowest common multiple of 4 and 6 is 12, so 11/4 = 33/12 and 11/6 = 22/12. Adding gives 55/12, or 4.583333. Back as a mixed number: 12 goes into 55 four times with 7 left over, so 4 7/12.
Edge case: an answer that is a whole number
7/10 + 3/10 needs no rescaling: 7 + 3 = 10, giving 10/10, which reduces to 1. The calculator on this site prints a bare 1 rather than 1/1, because a denominator of one adds nothing.
Edge case: a negative fraction
Adding a negative is subtracting. 1/2 + (−3/4) uses the common denominator 4: 2/4 − 3/4 = −1/4, or −0.250000. Keep the minus sign with the numerator while you work, and remember that −3/4, 3/−4 and −(3/4) are all the same number.
| Sum | Common denominator | Rescaled | Answer | Decimal |
|---|---|---|---|---|
| 1/2 + 1/3 | 6 | 3/6 + 2/6 | 5/6 | 0.833333 |
| 3/8 + 5/12 | 24 | 9/24 + 10/24 | 19/24 | 0.791667 |
| 1/6 + 1/8 | 24 | 4/24 + 3/24 | 7/24 | 0.291667 |
| 1/4 + 1/6 | 12 | 3/12 + 2/12 | 5/12 | 0.416667 |
| 2/3 + 5/9 | 9 | 6/9 + 5/9 | 11/9 | 1.222222 |
| 11/4 + 11/6 | 12 | 33/12 + 22/12 | 55/12 | 4.583333 |
| 7/10 + 3/10 | 10 | 7/10 + 3/10 | 1 | 1.000000 |
| 1/2 + (−3/4) | 4 | 2/4 − 3/4 | −1/4 | −0.250000 |
Finding the lowest common denominator quickly
For small numbers, list the multiples of the larger denominator until one of them divides by the smaller: for 8 and 12, that is 12, 24 — stop, 24 ÷ 8 = 3. For larger numbers, use the rule that the lowest common multiple equals the product divided by the greatest common divisor. With 18 and 24, the greatest common divisor is 6, so the lowest common multiple is 18 × 24 ÷ 6 = 72.
If one denominator is a multiple of the other, you are already done: for 1/4 + 3/20, the common denominator is 20, so 5/20 + 3/20 = 8/20 = 2/5.
Common mistakes
- Adding the denominators. 1/2 + 1/3 = 2/5 is the single most frequent error. The denominator is a label for the piece size, not a quantity being pooled.
- Rescaling only the numerator. Turning 1/3 into 2/3 while aiming for sixths changes the value. Whatever you multiply the bottom by, multiply the top by as well.
- Leaving the whole-number part behind. With mixed numbers, 2¾ + 1⅚ is not 3 + (3/4 + 5/6). That route works, but only if you carry the extra whole number when the fractional part exceeds 1: 3/4 + 5/6 = 19/12 = 1 7/12, so the total is 4 7/12, not 3 7/12.
- Forgetting to reduce. 76/96 is a correct answer to 3/8 + 5/12, but marked work and comparisons usually expect 19/24.
- Trusting a rounded decimal. 1/3 + 1/3 is exactly 2/3, not 0.666666. Add in fraction form when the result feeds into more arithmetic.
- A denominator of zero. 1/0 is not a very large number; it is undefined, and so is dividing by a fraction whose value is zero.
Check it yourself
Enter the two fractions in the fraction calculator, choose Addition and select Calculate. It keeps exact whole-number arithmetic throughout, reduces the result by the greatest common divisor, and shows a decimal rounded to six places alongside it. For 3/8 and 5/12 it returns 19/24 and 0.791667.
Three input habits worth knowing. Mixed notation such as 1 3/4 is not accepted, so enter 7/4. Decimals are accepted and converted exactly, so 0.1 plus 0.2 returns 3/10 rather than a binary rounding artefact. The minus sign can sit on either the numerator or the denominator. Because the decimal is only a display, a tiny non-zero answer can show as 0.000000 — read the fraction when precision matters.
Related tools and guides
Reviewed October 1, 2026. Calculation methods and corrections.