Fraction Calculator — Add, Subtract, Multiply, Divide

One third cannot be written exactly as a decimal

This is why fractions survive in engineering, cooking and music rather than being a school relic. Written as 0.333, one third is simply wrong — the true value runs on forever.

The error compounds through a calculation. 1/3 + 1/3 + 1/3 is exactly 1. But 0.333 + 0.333 + 0.333 is 0.999. In casual use the gap vanishes; in a structural calculation, a scaled recipe or a machined part, it does not.

Computers cannot store 0.1 exactly either

You may have noticed that in most programming languages 0.1 + 0.2 does not equal 0.3. This is not a language bug — it is the same problem in a different base.

Computers store decimals in binary, and just as one third has no exact representation in base ten, one tenth has none in base two. Keeping values as exact fractions sidesteps the entire class of error, which is why money systems generally work in whole sub-units — cents, pence, paise — rather than in decimal fractions of the main unit.

The mechanics, and why each rule exists

Addition and subtraction need a common denominator, because you cannot add parts of different sizes. Thirds and quarters are not the same unit, so both must be converted to twelfths first: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

Multiplication needs no common denominator, because multiplying scales rather than combines. Numerators multiply and denominators multiply: 2/3 × 3/4 = 6/12, which reduces to 1/2.

Division inverts the second fraction, because dividing by a half asks how many halves fit inside. Three contains six halves, so 3 ÷ 1/2 = 3 × 2/1 = 6 — dividing by a number below one makes the result larger, which stops being surprising once you read it as a question about how many fit.

Results reduce to lowest terms

Every answer is reduced automatically by dividing both parts by their greatest common divisor, so 6/8 becomes 3/4. The value is unchanged — only the representation gets simpler, since the two are the same number written differently.

Each result also appears as a decimal, to six places with trailing zeros trimmed, and as a mixed number where the fraction is improper: 7/3 shows as 2 1/3. The decimal is there for convenience; where it terminates it is exact, and where it does not — as with thirds and sevenths — the fraction beside it is the accurate one.

Where fractions are still the working unit

In each of these the fraction is not a tidied-up decimal, it is the actual quantity:

• Construction and imperial measurement work in halves down to sixteenths. A blueprint says 3/8 inch, not 0.375, because the tooling and the tape measure are divided that way.
• Recipes scale cleanly by fractions — doubling 1/3 cup gives 2/3 cup, and a measuring cup has that marking. It has no marking for 0.667.
• Musical time signatures are fractions by definition: 4/4, 3/4, 6/8.
• Gear and pulley ratios are inherently fractional, since they count teeth — whole numbers on both sides.

A fraction and a percentage are the same quantity written two ways, and converting between them is where most arithmetic slips happen. The percentage calculator handles that side.