Key facts
3/4,0.75and75%are three ways of writing the same quantity.- Fraction to decimal is one division: numerator ÷ denominator.
- A fraction in lowest terms terminates exactly only when its denominator has no prime factors other than 2 and 5.
- NumberUtils converts a decimal exactly as typed, so
0.333becomes333/1000, never1/3.
One quantity, three forms
Every fraction can be written as a decimal, and every decimal as a percentage. Percent means per hundred, so 75% is the fraction 75/100, which simplifies to 3/4 and has the decimal value 0.75.
Conversion between the three forms is exact whenever the answer can be written out in full. It stops being exact only where a recurring decimal has to be cut short, and that is where rounding errors begin.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.333…% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 2/3 | 0.666… | 66.666…% |
| 3/4 | 0.75 | 75% |
| 1/10 | 0.1 | 10% |
| 1/100 | 0.01 | 1% |
Values written with an ellipsis carry on for ever. 1/3 has no exact decimal form, so treat any decimal for it as a rounded figure.
Converting a fraction to a decimal
Divide the numerator by the denominator. Three quarters is 3 ÷ 4 = 0.75, and the division finishes there, so 0.75 is the exact value rather than an approximation.
One third does not finish: 1 ÷ 3 runs on as 0.333333333333… The fraction to decimal calculator shows 0.3333333333, the value rounded to 10 decimal places. Numerators and denominators must be whole numbers, and the denominator cannot be zero.
Converting a decimal to a fraction
A decimal is a fraction whose denominator is a power of ten: 0.1 is 1/10 and 0.125 is 125/1000. To convert one, count the digits after the decimal point, write those digits as a whole number over the matching power of ten, then reduce.
- 0.75 has two decimal places, so it is 75/100. Both parts divide by 25 to give 3/4.
- 0.125 has three decimal places, so it is 125/1000. Both parts divide by 125 to give 1/8.
The decimal to fraction calculator performs exactly that scaling, using the number of decimal places you typed. A finite decimal therefore always becomes the precise fraction it stands for, never a nearby approximation.
Converting a percentage to a fraction, and back
Divide the percentage by 100 to reach a fraction: 12.5% is 12.5/100, which is 125/1000, which reduces to 1/8.
To go the other way, multiply the fraction by 100. One eighth is 0.125, and 0.125 × 100 = 12.5%. The percentage to fraction calculator and the fraction to percentage calculator both hold the value as a fraction until the final decimal step, so 1/8 comes back as exactly 12.5%.
Simplifying with the greatest common divisor
A fraction is in its simplest form when its numerator and denominator share no factor other than 1. To simplify, divide both parts by their greatest common divisor.
For 18/24 that divisor is 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4. Finding the shared factors of a number is covered in the factors and primes guide, and the GCD and LCM calculator returns the greatest common divisor of two numbers directly.
Terminating and recurring decimals
A fraction in lowest terms has a terminating decimal exactly when its denominator is built only from 2s and 5s, because 10 is 2 × 5. The number of decimal places it needs is then the larger of the two counts of those factors.
- 1/8 has denominator 8, made of 2s only, so it terminates as 0.125 in three decimal places.
- 1/3 has denominator 3, so it recurs as 0.333333…
- 1/6 has denominator 6 = 2 × 3, so it recurs as 0.166666…
- 1/7 has denominator 7, so it recurs as 0.142857…
Apply the test to the fraction in lowest terms. The denominator of 3/12 contains a 3, but 3/12 reduces to 1/4, which terminates as 0.25.
Why 0.333 is not one third
NumberUtils reads a decimal exactly as entered, so 0.333 is 333/1000. Those two parts share no factor and nothing can be cancelled: 0.333 is a genuinely different number from 1/3, smaller by exactly 1/3000.
Multiply 0.333 by three and the result is 0.999, not 1, and three rounded thirds will always leave a residue of that kind. Where you mean one third exactly, enter the fraction rather than a rounded decimal, or state the precision being used, such as 33.3% to one decimal place.
Mixed numbers and improper fractions
A mixed number adds a whole number to a proper fraction, as in 2 3/4. To rewrite it as a single improper fraction, multiply the whole number by the denominator and add the numerator: (2 × 4) + 3 = 11, so 2 3/4 is 11/4.
To convert back, divide the numerator by the denominator. The quotient is the whole part and the remainder becomes the new numerator, so 11/4 is 2 with a remainder of 3.
A negative mixed number carries its sign on the whole number: −2 3/4 is −11/4, the negative of two and three quarters, not minus two plus three quarters. A fraction itself may also be negative, so −3/4 is an ordinary value rather than an error.
Adding fractions
Fractions can only be added when they share a denominator. One half plus one third mixes halves and thirds, so rewrite the half as 3/6 and the third as 2/6. The answer is 3/6 + 2/6 = 5/6.
The fraction calculator handles both parts of this, including mixed numbers and negative values.
Multiplying and dividing
Multiplying means multiplying the numerators together and the denominators together, so 1/2 × 1/3 = 1/6. Dividing means turning the second fraction upside down and multiplying, so 1/2 ÷ 1/3 is 1/2 × 3/1, which is 3/2.
Why staying exact matters
Fractions are kept as whole-number pairs while a calculation runs, so 1/3 + 1/3 + 1/3 comes out as exactly 1 rather than 0.9999999999999999. A decimal is produced only when one is asked for at the end. The fractions section of the methodology sets out how that is done.