Fraction to Percent Calculator
Result
As a percentage (%)
- Fraction in lowest terms
- 3/8
- Repeating digits (0 = terminating)
- 0
A fraction to percent calculator does two things at once, and the second one is easy to miss: it turns the fraction into a percentage, and it tells you whether the percentage it printed is the exact value or a rounded one. The conversion itself is the division the fraction already stands for, followed by one multiplication. Three eighths means three divided by eight, which is 0.375, and multiplying by a hundred gives 37.5 percent. The panel also does the step that would otherwise be done on paper: simplify the fraction into lowest terms and print it beside the result, so a fraction typed as 2/4 comes back as 1/2 — the same number, written the way it would be written by hand, which is also the form the other two readings are computed from. The third reading is where this page parts company with a plain division. It counts the repeating digits in the fraction's decimal expansion, and a zero there means the decimal terminates. That is not the same statement as the percentage being exact, and the gap between the two is worth stating plainly, because it is the thing most likely to be misread. A percentage is the fraction multiplied by a hundred, so it lands inside four decimal places only when the reduced denominator divides a million — that is, when it is built from no more than six twos and six fives. One over one hundred and twenty-eight is the case that shows the seam. Its denominator is two to the seventh, so its decimal is 0.0078125 and terminates, while its percentage is 0.78125 and needs a fifth place, so the panel prints 0.7813. The repeating digits are zero and the printed percentage is still rounded, both at once, and neither is a mistake. When that happens the exact percentage is still available by hand: read the fraction as the decimal it stands for and move the point two places to the right, or divide the denominator into the numerator after multiplying by a hundred. Three more things are worth knowing before you read the panel. Negative fractions are accepted and the sign sits on the numerator, so minus three eighths is minus 37.5 percent, and the repeating digits are counted on the absolute value of the denominator, which means the sign never changes them. Values above a hundred percent are accepted too and simply mean the fraction is above one, which happens whenever the numerator is larger than the denominator. And the denominator is capped at one hundred thousand, not for any mathematical reason but because the repeating digits are found by walking through the remainders one at a time, and a denominator near a prime with seven digits would walk a billion steps.
One divided by two through twelve
| Denominator | Percent | Repeating digits |
|---|---|---|
| 2 | 50 | 0 |
| 3 | 33.3333 | 1 |
| 4 | 25 | 0 |
| 5 | 20 | 0 |
| 6 | 16.6667 | 1 |
| 7 | 14.2857 | 6 |
| 8 | 12.5 | 0 |
| 9 | 11.1111 | 1 |
| 10 | 10 | 0 |
| 11 | 9.0909 | 2 |
| 12 | 8.3333 | 1 |
Each row is one over the denominator in the first column, so the numerator is one throughout and the row shows what the denominator alone does to the answer. The rows run in order of denominator rather than of size, because what the page teaches is a property of the denominator: a terminating decimal needs a denominator built only from twos and fives, and the rows where the repeating digits read zero are exactly the denominators that are. Only one of those rows needs a decimal place in its percentage: one eighth is 12.5 percent, while one half, one quarter, one fifth and one tenth come out as whole numbers. Read down the middle column and the repeating rows are the ones that go on past four places, which is why their printed percentages are rounded and the others are not. The denominator that shows the seam between the two answers is not here, because it is one hundred and twenty-eight and this table stops at twelve.
Formula
Percent = numerator ÷ denominator × 100
- Numerator
- The top number, and the one that carries the sign. It may be larger than the denominator, which gives a percentage above a hundred, and it may be zero, which gives zero.
- Denominator
- The bottom number, a positive integer up to one hundred thousand. It may not be zero, and it may not be a decimal: a fraction of decimals is a different problem, solved by writing it as a fraction of integers first.
- Percent
- The fraction multiplied by a hundred, printed to four decimal places. Whether those four places hold the exact value depends on the denominator, and the repeating digits below are the first half of that answer rather than the whole of it.
- Lowest terms
- The same fraction with the common factors cancelled, which is the form the other two readings come from. The repeating digits of a decimal do not change when a fraction is reduced, but whether the percentage is exact does depend on this form.
- Repeating digits
- How many digits of the decimal repeat forever, with zero meaning the decimal terminates. It is a count of digits and not the digits themselves: one sixth repeats a single digit and reads one, while one eleventh repeats two and reads two.
Three uses are worth separating. The first is a score or a share that arrives as a fraction and has to be compared with something given as a percentage: twenty-seven out of forty is 67.5 percent, and knowing that the printed figure is exact matters when it is going into a report. The second is a fraction that has to be simplified at the same time as it is converted, which is what the second reading is for. The third is a fraction whose percentage is not exact, where the question is how far off the printed figure is: one third prints as 33.3333 percent, which is a third of a thousandth of a percent low, and that is worth knowing before quoting it to two decimal places.
Worked examples
Three eighths as a percentage
- Divide: 3 ÷ 8 = 0.375
- Multiply by a hundred: 0.375 × 100 = 37.5
- Check the decimal: 8 is two cubed, so it terminates and the repeating digits read 0
The default case, and one where all three readings agree: the fraction is already in lowest terms, the decimal terminates and the percentage fits in two of its four places.
One third, where the percentage cannot be exact
- Divide: 1 ÷ 3 = 0.333333…
- Multiply by a hundred: 33.33333… percent
- Round to four places: 33.3333, and the repeating digits read 1
The repeating digits say one digit repeats, which is the whole of the reason the printed percentage is short of the true value: 33.3333 is a third of a thousandth of a percent below one third of a hundred.
Two quarters, which the panel simplifies
- Reduce first: 2/4 = 1/2
- Divide: 1 ÷ 2 = 0.5
- Multiply by a hundred: 50 percent, and the repeating digits read 0
The fraction on the panel is not the fraction that went in. Two quarters and one half are the same number, and the readings are computed from the reduced form, which is the form the denominator rule needs to be read against.
One over one hundred and twenty-eight, where the two answers come apart
- Divide: 1 ÷ 128 = 0.0078125, which terminates, so the repeating digits read 0
- Multiply by a hundred: 0.78125 percent
- That is five decimal places and the panel holds four, so it prints 0.7813
This is the case the page exists to warn about. A zero in the repeating digits means the decimal terminates, not that the printed percentage is exact. Here it is rounded, by exactly half of the last place it shows.
Limitations
The percentage is printed to four decimal places, so a fraction whose percentage needs more places is rounded rather than refused, and the panel gives no flag saying so. The repeating digits answer a different question: zero means the decimal terminates, and a denominator holding more than six twos or more than six fives can terminate yet still need more than four places as a percentage. One over one hundred and twenty-eight is the smallest example, and one over two hundred and fifty-six, one over five hundred and twelve and one over six hundred and twenty-five behave the same way. The denominator is limited to one hundred thousand, which is a limit on how long the repeating digits are allowed to take to find rather than a limit on what a fraction means; a fraction with a larger denominator is refused rather than answered slowly. The fraction must be whole numbers, so a fraction written with decimals in it has to be turned into one with integers before it arrives here. And the page does not add the two readings together into a verdict: it will print a rounded percentage without telling you the size of the rounding, which for one third is a third of a thousandth of a percent and for one over one hundred and twenty-eight is half of the last place shown.
Frequently asked questions
- How do I convert a fraction to a percentage?
- Divide the numerator by the denominator and multiply by a hundred. Three eighths is three divided by eight, which is 0.375, and that is 37.5 percent. Multiplying by a hundred is the same as moving the decimal point two places to the right.
- Why does the panel show the fraction in lowest terms?
- Because the other two readings are computed from that form, and because the rule for whether a decimal repeats is a statement about the reduced denominator rather than the one that was typed. Two quarters comes back as one half, and so does four eighths.
- What does the repeating digits number mean?
- It is how many digits of the fraction's decimal go on repeating forever, and zero means the decimal stops. One half reads zero, one third reads one because a single digit repeats, and one eleventh reads two because a pair repeats. It is a count of digits, not the digits themselves.
- Can the percentage be rounded when the repeating digits are zero?
- Yes, and that is the case worth knowing about. A zero means the decimal terminates. The percentage is the decimal times a hundred, so it needs two more places than the decimal did, and a fraction like one over one hundred and twenty-eight terminates at seven places and prints as 0.7813 percent because only four are shown. The exact value is 0.78125, half of the last place up.
- Why is the percentage shown to four decimal places?
- So that percentages from different calculators on this site can be compared directly. The same four places are used by the decimal to percent converter and by the basis point converter, which means a figure copied from one page can be checked against another without first asking how many places each one keeps.
- Can the answer be more than a hundred percent?
- Yes, whenever the numerator is larger than the denominator. Seven over one is seven hundred percent, and five over four is a hundred and twenty-five percent. Nothing here treats a fraction above one as an error.
- What if the fraction is negative?
- The sign sits on the numerator, so minus three eighths is minus 37.5 percent and the fraction prints as -3/8. The repeating digits are found from the size of the denominator, so the sign never changes them.
References
- Reduced Fraction — a fraction written with the common factors of numerator and denominator cancelled, which is the form the denominator rule has to be applied to — Wolfram MathWorld (United States)
- Repeating Decimal — a decimal expansion in which a digit or block of digits repeats forever, together with the rule that a terminating decimal is one whose reduced denominator has no prime factor other than 2 or 5 — Wolfram MathWorld (United States)
- Percent — a ratio or fraction is converted to a percentage by multiplying by a hundred and appending the percentage sign, which is the second of the two steps this page performs — Wolfram MathWorld (United States)