How to add fractions: same and different denominators, mixed numbers and simplifying
To add fractions with the same denominator, add the numerators and keep the denominator: 2/7 + 3/7 = 5/7. If the denominators differ, first rewrite both fractions over a common denominator, best the least common multiple of the denominators, then add the numerators and simplify: 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
Adding fractions comes down to one idea: you can only add pieces of the same size. This guide shows the rule for equal denominators, the least common denominator for different ones, mixed numbers and subtraction, and how to simplify an answer with the greatest common divisor, with a worked example at every step. At the end, a free calculator checks your work and writes out each step.
Key ideas
- Same denominator: add the numerators and keep the denominator, so 3/8 + 1/8 = 4/8 = 1/2. Never add the denominators.
- Different denominators: rewrite both fractions over the least common denominator, the LCM of the denominators, then add: 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
- Mixed numbers become improper fractions first: 2 3/4 = (2 × 4 + 3)/4 = 11/4. Subtraction follows the same steps with a minus sign.
- To simplify, divide the numerator and the denominator by their greatest common divisor: 63/462 ÷ 21 gives 3/22, in lowest terms.
How do you add fractions with the same denominator?
Add the numerators and keep the denominator. The denominator names the size of the pieces and the numerator counts them, so two quarters plus three quarters are five quarters: 2/4 + 3/4 = 5/4. Only like quantities can be added, which is why the pieces have to be the same size before you count them together.
A worked example: 3/8 + 1/8. The denominators are both 8, so add 3 + 1 = 4 and keep the eighths: 4/8. That answer is correct but not finished, because 4/8 can be simplified to 1/2. The most common mistake is to add the denominators as well. It cannot be right: 1/2 + 1/2 would then give 2/4, which is still a half, not a whole.
Subtraction works the same way: 7/9 − 4/9 = 3/9, which simplifies to 1/3. When the numerator ends up larger than the denominator, the result is an improper fraction, and you may write it as a mixed number: 5/4 is 1 1/4, because 5 divided by 4 is 1 with a remainder of 1.
How do you add fractions with different denominators?
Rewrite both fractions over a common denominator, then add the numerators. The smallest common denominator is the least common multiple (LCM) of the denominators, called the least common denominator. First find the LCM, then multiply the top and bottom of each fraction by whatever turns its denominator into the LCM, then add the numerators, and finally simplify.
Take 1/4 + 1/6. The multiples of 4 are 4, 8, 12…, the multiples of 6 are 6, 12…, so the LCM is 12. Multiply 1/4 by 3/3 to get 3/12 and 1/6 by 2/2 to get 2/12. Now 3/12 + 2/12 = 5/12. A harder pair: 5/12 + 11/18. The least common multiple of 12 and 18 is 36, so the sum is 15/36 + 22/36 = 37/36, or 1 1/36.
Multiplying the two denominators always gives a common denominator too: 1/4 + 1/3 = 3/12 + 4/12 = 7/12. It just may not be the smallest. For 3/4 + 5/6 the product is 24, giving 18/24 + 20/24 = 38/24 = 19/12, while the LCM 12 gives 9/12 + 10/12 = 19/12 straight away. Subtracting is the same: 5/6 − 1/4 = 10/12 − 3/12 = 7/12.
How do you add and subtract mixed numbers?
Turn each mixed number into an improper fraction, add or subtract as usual, then turn the answer back. To convert, multiply the whole part by the denominator and add the numerator: 2 3/4 = (2 × 4 + 3)/4 = 11/4.
Example: 1 1/2 + 2 2/3. Converted, that is 3/2 + 8/3. The LCM of 2 and 3 is 6, so 9/6 + 16/6 = 25/6, and 25 ÷ 6 = 4 remainder 1 gives 4 1/6. You can also add the whole parts and the fractions separately: 1 + 2 = 3 and 1/2 + 2/3 = 7/6 = 1 1/6, so the total is again 4 1/6.
Subtraction: 3 1/4 − 1 2/3 = 13/4 − 5/3 = 39/12 − 20/12 = 19/12 = 1 7/12. Done part by part, this one needs borrowing, because 1/4 is smaller than 2/3: take 1 from the 3 to make 2 5/4, then 5/4 − 2/3 = 15/12 − 8/12 = 7/12 and 2 − 1 = 1, which gives the same 1 7/12.
How do you simplify fractions, and how can you check the answer?
Divide the numerator and the denominator by their greatest common divisor (GCD); the result is the same number in lowest terms. For 6/8 the GCD is 2, so 6/8 = 3/4. For 63/462 the GCD is 21, so it becomes 3/22. A fraction is fully simplified when the GCD of its two parts is 1, as in 5/12.
For large numbers, find the GCD with Euclid's algorithm: divide the larger number by the smaller, keep the remainder, and repeat with the smaller number and that remainder until the remainder is 0. The last non-zero remainder is the GCD. For 462 and 63: 462 = 7 × 63 + 21, then 63 = 3 × 21 + 0, so the GCD is 21. With small numbers you can also divide by common factors one at a time: 18/24 → 9/12 → 3/4.
To check homework, open the free fraction calculator at VITON13 SCHOOL. Enter up to four numbers, each a whole number, a fraction or a mixed number, with + − × ÷ between them, or switch to Expression and type 1 1/2 + 2 2/3. The answer appears as a simplified fraction, a mixed number, a decimal and a percent, with each step written out: mixed numbers converted, the lowest common denominator, the rewritten fractions and simplifying by the GCD. The Simplify mode shows Euclid's algorithm for one fraction, and bars or circles picture the sizes.
In everyday language
Fractions are like slices of pizza: you can only count slices together when they are the same size. Quarters and sixths first have to be cut into twelfths; after that you simply count the twelfths.
Try it yourself
Add 2/3 + 3/4 on paper using the least common denominator, then enter the same sum in the fraction calculator and compare its steps with yours.
Expected result
The common denominator 12, the fractions 8/12 and 9/12, the sum 17/12 and the mixed number 1 5/12.
Check your answer: What is 3/10 + 1/6 in lowest terms?
7/15. The LCM of 10 and 6 is 30, so 3/10 = 9/30 and 1/6 = 5/30. Then 9/30 + 5/30 = 14/30, and dividing both parts by the GCD, 2, gives 7/15.
Questions
Do you add the denominators when adding fractions?
No. The denominator names the size of the pieces, so once the fractions share one it stays the same and only the numerators are added. Adding denominators would make 1/2 + 1/2 equal 2/4, which is a half rather than a whole.
Do I have to use the least common denominator?
No, any common denominator works, for example the product of the two denominators. The least one keeps the numbers small, and whichever you choose, the simplified answer is the same.
Same denominator: add the numerators. Different denominators: find the LCM, rewrite, add, then simplify by the GCD. Work a few sums on paper and check each one in the free VITON13 SCHOOL fraction calculator, which writes every step out.
Sources and further reading
- Wikipedia: Fraction (arithmetic with fractions) ↗Sources checked:
- Wikipedia: Lowest common denominator ↗Sources checked:
- Wikipedia: Euclidean algorithm ↗Sources checked: