A half plus a third
Halves and thirds do not line up, so both are rewritten over 6: 3/6 + 2/6.
That gives 5/6, which is already in lowest terms — about 0.833. Notice it is less than one, which rules out the common mistake of adding the denominators.
Fraction calculator
Calculate with two fractions, reduce one to its simplest form, or find the fraction behind a decimal — each with the working laid out step by step.
Result
Fill in the fields to see your result.
How it works
Addition and subtraction need the two fractions to be counting the same thing. Thirds and halves cannot be added directly, so both are rewritten over a common denominator — the lowest common multiple of the two, which keeps the numbers as small as possible. Once the denominators match, only the numerators move.
Multiplication needs no such preparation: numerators multiply together, denominators multiply together, and that is the whole rule. Division is multiplication in disguise — turn the second fraction over and multiply. That is why dividing by a half doubles a number rather than halving it.
Every answer is then reduced by dividing both parts by their greatest common divisor. A fraction in lowest terms is the one form mathematicians agree on, which is why 2/4, 3/6 and 50/100 all end up as 1/2.
Formulas used
Adding and subtracting
a/b ± c/d = (a × d ± c × b) ÷ (b × d)
Using the lowest common multiple of b and d instead of b × d keeps the numbers smaller.
Multiplying
a/b × c/d = (a × c) ÷ (b × d)
Dividing
a/b ÷ c/d = (a × d) ÷ (b × c)
The second fraction is inverted, then multiplied.
Worked examples
Simplifying 18/24
The greatest common divisor of 18 and 24 is 6. Dividing both parts by it gives 3/4.
The value never changed — 18/24 and 3/4 are the same number, 0.75. Only the way of writing it got simpler.
What fraction is 0.375?
Exactly 3/8. The decimal stops after three places, which is the signature of a denominator built only from 2s.
Try 0.333333 instead and the answer becomes 1/3 — close, but not exact, because that decimal never truly ends.
Assumptions and limits
- Numerators and denominators are whole numbers.
- A negative whole part applies to the entire mixed number.
- Results are reduced to lowest terms.
- Decimal conversions are limited by the largest denominator you allow.
Frequently asked questions
Why do I need a common denominator to add but not to multiply?
Because addition counts parts, and you can only count parts of the same size. Two thirds plus one half is not three of anything until both are cut into sixths.
Multiplication asks a different question — a fraction of a fraction — and the sizes take care of themselves. Half of a third is a sixth, and the denominators multiplying together is exactly why.
What is an improper fraction?
One where the numerator is larger than the denominator, such as 7/4. Nothing is wrong with it; it is simply more than a whole.
Written as a mixed number it becomes 1 3/4. The calculator shows both, because arithmetic is easier with improper fractions and reading is easier with mixed numbers.
Which fractions turn into repeating decimals?
Those whose denominator, once reduced, contains a prime factor other than 2 or 5. One third and one seventh repeat; one eighth and three twentieths stop.
The reason is that our decimal system is built on ten, and ten factors into two and five. A denominator with anything else in it never divides evenly into a power of ten.
How do I divide by a fraction?
Turn it upside down and multiply. Dividing by 1/2 is multiplying by 2, which is why "how many halves fit in 3" gives 6.
The step is shown explicitly in the working, since it is the one rule most people remember as a slogan without knowing why it holds.
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