Last updated 3 September 2026
How fraction arithmetic works
- Adding / subtracting — cross-multiply to a common denominator first: a/b ± c/d = (ad ± bc) / bd.
- Multiplying — straightforward: multiply the numerators together and the denominators together.
- Dividing — flip the second fraction and multiply: a/b ÷ c/d = a/b × d/c.
Every result is reduced to its simplest form by dividing both the numerator and denominator by their greatest common divisor, so 6/8 comes back as 3/4, never left unreduced.
Why addition needs a common denominator
Multiplying fractions is straightforward — multiply the tops, multiply the bottoms, and 2/3 × 3/4 gives 6/12, or 1/2. Addition is where the rules stop being intuitive, because a fraction is a count of parts and you can only add counts of parts that are the same size.
So 1/3 + 1/4 cannot be 2/7. Thirds and quarters are different-sized pieces. Rewriting them over a shared denominator of 12 gives 4/12 + 3/12 = 7/12, which is the correct answer and slightly more than a half. The 2/7 answer is not just wrong, it is smaller than either of the two fractions being added, which is a useful sanity check to carry around.
A worked example from a recipe
A recipe serving four needs 2/3 cup of flour, and you are cooking for six. Six is 3/2 of four, so you need 2/3 × 3/2 = 6/6 = 1 cup exactly. Scaling the same recipe down for three people means multiplying by 3/4: 2/3 × 3/4 = 6/12 = 1/2 cup.
Mixed numbers are the other common stumbling point. To work with 2 1/4, convert it to an improper fraction first — multiply the whole number by the denominator and add the numerator, giving 9/4. Doing arithmetic on the whole part and the fraction part separately works for addition but breaks immediately for multiplication and division.
Common questions
How do you add fractions with different denominators?
Cross-multiply so both fractions share a common denominator, add the numerators, then simplify. For a/b + c/d, that's (ad + bc) / (bd), reduced by dividing both parts by their greatest common divisor.
Why does the result get automatically simplified?
6/8 and 3/4 are the same value, but 3/4 is the form anyone would expect as an answer. The calculator divides both numerator and denominator by their greatest common divisor so the result is always in its simplest form.
What does the mixed-number result mean?
It's the same value written as a whole number plus a fraction, such as 7/4 shown as 1 3/4. It only appears when the result is greater than one whole.
Why can I not just add the tops and the bottoms?
Because a fraction counts parts of a particular size, and thirds are not the same size as quarters. Adding 1/3 and 1/4 as 2/7 produces an answer smaller than either original fraction, which is impossible. Rewrite both over a shared denominator first: 4/12 + 3/12 = 7/12.
How do I convert a mixed number like 2 1/4?
Multiply the whole number by the denominator, add the numerator, and keep the same denominator. So 2 1/4 becomes (2 × 4 + 1)/4 = 9/4. Convert before multiplying or dividing — handling the whole and fractional parts separately gives wrong answers for both operations.
Why is dividing by a fraction the same as multiplying by its reciprocal?
Dividing asks how many of the second fraction fit into the first. Asking how many halves fit into 3 is the same as asking what 3 × 2 is, which is why 3 ÷ 1/2 = 6. Flipping the divisor and multiplying is the shortcut for that question.