Fraction Calculator
Add, subtract, multiply, and divide fractions. Results are automatically simplified.
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What is Fractions?
A fraction is a division that has not been carried out yet. Three quarters means three divided by four, and every rule for manipulating fractions follows from that single fact rather than from arbitrary convention.
That framing resolves most of the confusion. Why do you need a common denominator to add but not to multiply? Why does dividing by a fraction flip it? Neither is a rule to memorise — both fall out of what division means, and both are derived below.
Fractions are also more precise than decimals, which is why they persist in engineering, cooking and music. One third is exact; 0.333 is not, and the error compounds through a calculation.
The formula — how to calculate Fractions
- Numerator (a, c)
- = the top — how many parts you have
- Denominator (b, d)
- = the bottom — how many parts make a whole
Multiplying by bd in the addition rule is just rewriting both fractions over a shared denominator. It always works, though it may not give the smallest one.
Step-by-step example
- 01Add 2/3 + 1/4.
- 02The pieces are different sizes, so they cannot be counted together yet. Rewrite both with denominator 12: 2/3 = 8/12 and 1/4 = 3/12.
- 03Now the pieces match: 8/12 + 3/12 = 11/12. Nothing divides both 11 and 12, so it is already simplest.
- 04Multiply 2/3 × 1/4.
- 05Two thirds OF one quarter: take a quarter, split it into three, keep two. The result is 2/12 = 1/6. Numerators multiply, denominators multiply — no common denominator needed, because you are not counting pieces together, you are taking a portion of a portion.
- 06Divide 2/3 ÷ 1/4.
- 07This asks: how many quarters fit into two thirds? Since four quarters fit in one whole, multiply by 4/1: (2 × 4)/(3 × 1) = 8/3 = 2⅔.
- 08That is why dividing flips the second fraction — dividing by a quarter is the same as multiplying by four.
Why addition needs a common denominator
The denominator names the size of the piece. Thirds and quarters are different-sized pieces, so "two thirds plus one quarter" is like asking how many objects you have if you own two apples and one orange — you cannot combine the counts until you describe both in the same unit.
Converting to twelfths gives both quantities the same unit: eight twelfths and three twelfths. Now they are the same kind of thing and the numerators simply add.
Multiplication needs no such conversion because it is not combining counts. Multiplying asks for a portion of a portion, and the sizes are supposed to change — that is the operation.
The lowest common denominator is the smallest number both denominators divide into. Using it keeps the numbers manageable, but multiplying the denominators together always works and simplifies afterwards.
Why dividing flips the fraction
Division asks "how many of these fit into that?". Dividing by 1/4 asks how many quarters fit, and since four fit into every whole, dividing by 1/4 multiplies by 4. Generally, dividing by a/b multiplies by b/a — the reciprocal. It is not a trick; it is what the question means.
Simplifying, and why it matters
A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. Simplifying does not change the value — 6/8 and 3/4 are the same number — it changes only how it is written.
The systematic method is to divide both parts by their greatest common divisor. For 18/24 that is 6, giving 3/4. Without spotting the GCD, repeated halving or dividing by any shared factor gets there eventually.
The practical reason to simplify is that unsimplified fractions compound. Multiplying several unsimplified fractions produces large numerators and denominators quickly, and errors hide in large numbers.
Common conversions
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/8 | 0.125 | 12.5% |
| 2/3 | 0.666… | 66.7% |
| 3/4 | 0.75 | 75% |
| 5/8 | 0.625 | 62.5% |
A fraction terminates as a decimal only if its simplified denominator has no prime factors other than 2 and 5. That is why halves, quarters and eighths are clean while thirds and sevenths repeat forever.
Mixed numbers and improper fractions
An improper fraction has a numerator at least as large as its denominator — 8/3. A mixed number writes the same value as a whole part plus a remainder — 2⅔.
Converting is division with a remainder: 8 ÷ 3 is 2 remainder 2, so 8/3 = 2⅔. Going back, multiply the whole number by the denominator and add the numerator: 2 × 3 + 2 = 8, so 2⅔ = 8/3.
Mixed numbers are easier to picture, which is why recipes and measurements use them. Improper fractions are easier to compute with, which is why you should convert to them before multiplying or dividing and convert back at the end.
Key considerations
- Convert mixed numbers to improper fractions before multiplying or dividing.
- Simplify as you go rather than at the end — the numbers stay smaller and errors are easier to spot.
- Any common denominator works; the lowest one just keeps arithmetic manageable.
- A fraction terminates as a decimal only when its denominator has just 2s and 5s as prime factors.
- Fractions are exact where decimals round, which matters in repeated calculations.
- A negative sign belongs to the fraction as a whole; −3/4, (−3)/4 and 3/(−4) are the same number.
Common mistakes to avoid
- Adding numerators and denominators separately — 1/2 + 1/3 is not 2/5.
- Looking for a common denominator when multiplying, where none is needed.
- Flipping the wrong fraction when dividing; it is the second one that inverts.
- Leaving an answer unsimplified so later steps carry unnecessarily large numbers.
- Multiplying mixed numbers without converting to improper fractions first.
- Rounding to a decimal early and carrying the error through the rest of the calculation.
Frequently asked questions
Sources & references
Written and fact-checked by the CalcProLabs Editorial Team. Read our calculation methodology and editorial policy.
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