Enter the decimal you want as a fraction and press "Calculate" to see it in simplest form. A decimal greater than 1 is also shown as a mixed number.
Enter a negative decimal like "-2.25". Repeating decimals (0.333…) are not supported (terminating decimals only).
Result
Enter a decimal in the field on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter a decimal such as 0.375, and you see it as a fraction in simplest form right away
- A decimal greater than 1, such as 1.375 → \(\dfrac{11}{8}\), is also shown as a mixed number (\(1\dfrac{3}{8}\))
- Negative decimals (for example, -2.25 → \(-\dfrac{9}{4}\)) and whole numbers are fine too. The steps "make the denominator a power of 10 → simplify with the greatest common factor" are shown as well
- A plain-language explanation of how to convert, and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page converts decimals that come to an end (terminating decimals) such as 0.375. It cannot turn a decimal whose digits repeat forever (a repeating decimal) such as 0.333… into \(\dfrac{1}{3}\). (Repeating decimal notation such as "0.3(3)" cannot be entered.)
What is this calculation used for?
Matching the fractions on measuring cups (cooking)
Recipes and measuring cups mostly use fractions such as "\(\dfrac{3}{4}\) cup". If a conversion chart or a digital scale gives you "0.75 cup", change it to \(0.75 = \dfrac{75}{100} = \dfrac{3}{4}\), and you can measure it with the \(\dfrac{3}{4}\) cup.
Being able to go back and forth between decimals and fractions makes baking from any recipe much easier.
Turning decimal inches into fractional inches (DIY and tools)
Bolts, pipes and wrenches are sized in fractional inches such as \(\dfrac{3}{8}\) inch. If your measurement is in millimeters, first change it to decimal inches (\(9.525 \div 25.4 = 0.375\)), then turn that into a fraction, \(0.375 = \dfrac{375}{1000} = \dfrac{3}{8}\), and pick the \(\dfrac{3}{8}\) inch tool. A caliper reading of 0.375 inch works the same way.
Decimal to fraction is a calculation you really need when choosing parts at the hardware store or working with tools from another measuring system.
Saying a proportion as "7 out of 20"
Hearing "the share in favor is 0.35" does not mean much to most people, but \(0.35 = \dfrac{35}{100} = \dfrac{7}{20}\) lets you say "7 out of 20 people are in favor".
In news, reports and presentations, turning a decimal proportion into a fraction such as "X out of Y" makes the same number much easier to understand.
Writing a scale as a fraction (models and maps)
Scales on models and maps are written as fractions or ratios such as \(\dfrac{1}{25}\) or 1:25. If a model is 0.04 times the size of the real car, change it to \(0.04 = \dfrac{4}{100} = \dfrac{1}{25}\), and it is a 1:25 scale model.
For models, maps and plans, a fraction shows "what fraction of the real size" much more directly than a decimal, so this conversion is used all the time.
Note lengths in music are fractions (music and recording software)
In written music, a whole note counts as 1, and the other notes are named as fractions of it: half note = \(\dfrac{1}{2}\), quarter note = \(\dfrac{1}{4}\), eighth note = \(\dfrac{1}{8}\). If music software shows a note length of "0.125" with a whole note as 1, then \(0.125 = \dfrac{125}{1000} = \dfrac{1}{8}\), so it is an eighth note (some software measures length in beats or other units instead).
Turning decimals into fractions lets you move between the numbers you type in and the language of written music.
Formula
The basic formula for turning a decimal into a fraction (make the denominator 10, 100, 1000…)
Standard notation
(the usual math form)
\(x\)
\(=\)
\(a\)
\(\div\)
\(10^{n}\)
In words
(symbols replaced with words)
①
\(x\): decimal to convert
\(=\)
③
\(a\): digits without the decimal point
\(\div\)
②
\(10^{n}\): power-of-10 denominator
The formula in words
①
Count the number of decimal places \(n\) in the
\(x\): decimal to convert
②
put the
\(10^{n}\): power-of-10 denominator
(10 multiplied \(n\) times, that is, a 1 followed by \(n\) zeros) on the bottom
③
and put the
\(a\): digits without the decimal point
on top. The fraction \(\dfrac{a}{10^{n}}\) has the same value as the original decimal
Quick example
Turning 0.375 into a fraction (it has 3 decimal places, so the denominator is 10 multiplied 3 times, 1000) gives
decimal (0.375)
\(=\)
digits without the decimal point (375)
\(\div\)
power-of-10 denominator (1000)
\(0.375 = \dfrac{0.375 \times 1000}{1 \times 1000} = \dfrac{375}{1000}\)
Key idea
The first place after the decimal point is the tenths (\(\dfrac{1}{10}\)) place, and the second is the hundredths (\(\dfrac{1}{100}\)) place. In other words, a decimal is really a short way of writing a fraction whose denominator is 10, 100, 1000…. So if you put the power of 10 that matches the number of decimal places in the denominator, a decimal that comes to an end (a terminating decimal) can always be turned back into a fraction.
The rule at work is "multiplying the numerator and denominator by the same number does not change the value of a fraction". Think of \(x\) as \(\dfrac{x}{1}\) and multiply both parts by \(10^{n}\), and the decimal point in the numerator disappears.
The resulting fraction can often still be simplified, so simplify it next with formula 2.
Simplify to lowest terms (divide the numerator and denominator by the GCF)
Standard notation
(the usual math form)
\(N\)
\(=\)
\(a\)
\(\div\)
\(g\)
\(D\)
\(=\)
\(10^{n}\)
\(\div\)
\(g\)
In words
(symbols replaced with words)
③
\(N\): new numerator
\(=\)
①
\(a\): original numerator
\(\div\)
②
\(g\): greatest common factor
⑥
\(D\): new denominator
\(=\)
④
\(10^{n}\): original denominator
\(\div\)
⑤
\(g\): greatest common factor
The formula in words
①
Divide the
\(a\): original numerator
②
by the
\(g\): greatest common factor
(the largest number that divides both \(a\) and \(10^{n}\))
③
to get the
\(N\): new numerator
④
Then divide the
\(10^{n}\): original denominator
⑤
by the same
\(g\): greatest common factor
⑥
to get the
\(D\): new denominator
Quick example
Simplifying \(\dfrac{375}{1000}\) from formula 1 (the GCF of 375 and 1000 is 125) gives
new numerator (3)
\(=\)
original numerator (375)
\(\div\)
GCF (125)
new denominator (8)
\(=\)
original denominator (1000)
\(\div\)
GCF (125)
\(375 \div 125 = 3,\quad 1000 \div 125 = 8\)
\(0.375 = \dfrac{375}{1000} = \dfrac{3}{8}\)
Key idea
The greatest common factor is the largest of the numbers that divide both the numerator and the denominator (the common factors). It is also called the greatest common divisor (GCD), and you can get it in one step with the GCD function in Excel or math.gcd in Python.
When the denominator is a power of 10 (10, 100, 1000…), its only prime factors are 2 and 5. So this simplification is the same as "divide by 2 as many times as you can, and divide by 5 as many times as you can". If the numerator cannot be divided by 2 or 5, the fraction cannot be simplified any further (for example, 0.333333333333 → \(\dfrac{333333333333}{1000000000000}\) has a GCF of 1, so it stays as it is).
For a negative decimal, take off the minus sign, do the same calculation, and put the sign in front of the whole fraction at the end (for example, -2.25 → simplify \(\dfrac{225}{100}\) to \(\dfrac{9}{4}\) → \(-\dfrac{9}{4}\)).
Changing an improper fraction to a mixed number (for decimals greater than 1)
Standard notation
(the usual math form)
\(N\)
\(=\)
\(q\)
\(\times\)
\(D\)
\(+\)
\(r\)
In words
(symbols replaced with words)
①
\(N\): numerator of the improper fraction
\(=\)
③
\(q\): quotient (whole-number part)
\(\times\)
②
\(D\): denominator
\(+\)
④
\(r\): remainder (numerator of the fraction part)
The formula in words
①
Divide the
\(N\): numerator of the improper fraction
②
by the
\(D\): denominator
③
The
\(q\): quotient
is the whole-number part of the mixed number
④
and the
\(r\): remainder
is the numerator of the fraction part (the denominator stays \(D\))
Quick example
Changing \(\dfrac{11}{8}\), the improper fraction you get from 1.375 with formulas 1 and 2, to a mixed number (11 ÷ 8 = 1 remainder 3) gives
numerator of the improper fraction (11)
\(=\)
quotient (1)
\(\times\)
denominator (8)
\(+\)
remainder (3)
\(11 = 1 \times 8 + 3\)
\(1.375 = \dfrac{11}{8} = 1\dfrac{3}{8}\)
Key idea
When you turn a decimal greater than 1, such as 1.375, into a fraction, you get an improper fraction (numerator greater than or equal to the denominator) such as \(\dfrac{11}{8}\). Leaving it as an improper fraction is not wrong, but the mixed number \(1\dfrac{3}{8}\) ("one and three eighths") shows its rough size (a little more than 1) at a glance.
For a negative decimal, take off the minus sign, do the same calculation, and put the sign in front of the whole number at the end (for example, -2.25 → \(\dfrac{9}{4}\) = \(2\dfrac{1}{4}\) → with the minus sign, \(-2\dfrac{1}{4}\)).
To turn a decimal into a fraction, put a power of 10 with as many zeros as there are decimal places in the denominator, put the digits without the decimal point in the numerator, and simplify with the greatest common factor. For a decimal greater than 1, changing the improper fraction to a mixed number shows its rough size at a glance.
Symbols and terms
Symbols
| \(x\) | x | The original decimal you want as a fraction. (Examples - 0.375, 1.375, -2.25) |
| \(n\) | lowercase n | The number of decimal places (how many digits are to the right of the decimal point). (Example - for 0.375, \(n = 3\)) |
| \(10^{n}\) | 10 to the n | 10 multiplied by itself \(n\) times (10, 100, 1000, …). The small \(n\) at the upper right is the exponent, meaning "multiply \(n\) times", and it equals the number of zeros after the 1. On this page it goes in the denominator of the fraction. |
| \(a\) | a | The whole number you get by removing the decimal point from the original decimal. It goes in the numerator. (Example - 375 for 0.375) |
| \(g\) | g | The greatest common factor of the numerator and denominator. To simplify, divide both by this number. (Example - 125 for 375 and 1000) |
| \(\gcd(a,\ 10^{n})\) | GCD of a and 10 to the n | The notation for "the greatest common factor of \(a\) and \(10^{n}\)". GCD stands for greatest common divisor, another name for the greatest common factor. |
| \(N\) | capital N | The new numerator after simplifying. (Example - the 3 in \(\dfrac{3}{8}\), the simplified form of \(\dfrac{375}{1000}\)) |
| \(D\) | capital D | The new denominator after simplifying. (Example - the 8 in \(\dfrac{3}{8}\), the simplified form of \(\dfrac{375}{1000}\)) |
| \(q\) | q | The quotient when you divide the numerator by the denominator. It becomes the whole-number part of the mixed number. (From the first letter of "quotient") |
| \(r\) | r | The remainder when you divide the numerator by the denominator. It becomes the numerator of the fraction part of the mixed number. (From the first letter of "remainder") |
Terms
| terminating decimal | A decimal whose digits come to an end, such as 0.375. This calculator converts terminating decimals. |
| repeating decimal | A decimal in which the same block of digits repeats forever, such as 0.333…. The power-of-10 method on this page cannot convert it. It needs a different method (one that uses an equation), so this calculator does not handle it. |
| power of 10 | A number made by multiplying 10 by itself some number of times, such as 10, 100, 1000, …. The \(n\)th place after the decimal point is the "1 over \(10^{n}\)" place, so a power of 10 becomes the denominator when you turn a decimal into a fraction. |
| simplify | To divide the numerator and denominator of a fraction by a number that divides both (a common factor), making the fraction simpler. The value does not change. Also called reducing. |
| simplest form | A fraction that cannot be simplified any further (also called lowest terms). The greatest common factor of its numerator and denominator is 1. When a test says "write your answer in simplest form", this is the form it means. |
| greatest common factor (GCF) | The largest of the numbers that divide both of two numbers (their common factors). It is also called the greatest common divisor (GCD). |
| improper fraction | A fraction whose numerator is greater than or equal to its denominator, such as \(\dfrac{11}{8}\). A fraction worth 1 or more has this form. |
| mixed number | A number written as a whole-number part followed by a fraction part, such as \(1\dfrac{3}{8}\) ("one and three eighths"). It shows the same value as an improper fraction in a form where the size is easy to see. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.
| Decimal place value (Grades 4–5) |
|
| What a fraction means (Grade 3) |
|
| Simplifying fractions and common factors (Grades 4–6) |
|
| Mixed numbers, improper fractions and remainders (Grades 4–5) |
|
| Terminating and repeating decimals (Grades 7–8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to turn a decimal into a fraction over a power of 10
| Decimal to convert x | 1.375 |
| Decimal places n | 3 |
| Numerator a (no decimal point) | =B1*10^B2 |
| Denominator (power of 10) | =10^B2 |
Table to simplify to lowest terms
| Numerator a | 1375 |
| Denominator (power of 10) | 1000 |
| Greatest common factor g | =GCD(B1,B2) |
| New numerator N | =B1/B3 |
| New denominator D | =B2/B3 |
Table to change an improper fraction to a mixed number
| Numerator of the improper fraction N | 11 |
| Denominator D | 8 |
| Whole-number part (quotient q) | =QUOTIENT(B1,B2) |
| Numerator of the fraction part (remainder r) | =MOD(B1,B2) |
After pasting, the upper rows are your inputs, and the rows with formulas starting with "=" are calculated automatically.
"^" is a power (10^3 is 1000). GCD finds the greatest common factor, QUOTIENT the quotient (whole-number part), and MOD the remainder.
For 1.375, for example, the first table gives the numerator 1375 and denominator 1000 (B3 and B4), the second table simplifies with the GCF 125 to \(\dfrac{11}{8}\) (B4 and B5), and the third table gives the mixed number \(1\dfrac{3}{8}\) (B3 and B4). For a negative decimal, calculate with the value without its minus sign and add the sign at the end.
"^" is a power (10^3 is 1000). GCD finds the greatest common factor, QUOTIENT the quotient (whole-number part), and MOD the remainder.
For 1.375, for example, the first table gives the numerator 1375 and denominator 1000 (B3 and B4), the second table simplifies with the GCF 125 to \(\dfrac{11}{8}\) (B4 and B5), and the third table gives the mixed number \(1\dfrac{3}{8}\) (B3 and B4). For a negative decimal, calculate with the value without its minus sign and add the sign at the end.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to turn a decimal into a fraction over a power of 10
| Decimal to convert x | 1.375 |
| Decimal places n | 3 |
| Numerator a (no decimal point) | =B1*10^B2 |
| Denominator (power of 10) | =10^B2 |
Table to simplify to lowest terms
| Numerator a | 1375 |
| Denominator (power of 10) | 1000 |
| Greatest common factor g | =GCD(B1,B2) |
| New numerator N | =B1/B3 |
| New denominator D | =B2/B3 |
Table to change an improper fraction to a mixed number
| Numerator of the improper fraction N | 11 |
| Denominator D | 8 |
| Whole-number part (quotient q) | =QUOTIENT(B1,B2) |
| Numerator of the fraction part (remainder r) | =MOD(B1,B2) |
The same formulas as in Excel work as is (Google Sheets also has GCD, QUOTIENT and MOD). Copy the whole table, paste it into cell A1, and replace the inputs with your own decimal.
How to calculate it in Python
from fractions import Fraction
decimal_text = "1.375" # the decimal to convert (write it as a string to avoid rounding errors)
fraction_value = Fraction(decimal_text) # simplifying is done automatically
print(f"Fraction: {fraction_value}") # 11/8
# Change the improper fraction to a mixed number (whole-number part and remainder)
numerator = abs(fraction_value.numerator)
denominator = fraction_value.denominator
whole_part, remainder = divmod(numerator, denominator)
sign = "-" if fraction_value < 0 else ""
if whole_part > 0 and remainder > 0:
print(f"Mixed number: {sign}{whole_part} {remainder}/{denominator}")
The fractions module in the standard library does both steps for you: making the denominator a power of 10 and simplifying. If you pass a decimal (float) directly, as in Fraction(1.375), it may turn the tiny binary rounding errors inside the computer into the fraction too, so always pass it as a string such as "1.375". Change the decimal at the top and run it.
How to write it in LaTeX and other math languages (copy and paste)
The basic formula for turning a decimal into a fraction (make the denominator 10, 100, 1000…)
x = a ÷ 10ⁿ
x = \dfrac{a}{10^{n}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>a</mi>
<msup><mn>10</mn><mi>n</mi></msup>
</mfrac>
</mrow>
</math>
x = a / 10^n
a/10^n
x := a/10^n;
x = a/10^n;
x = a/10^n
Simplify to lowest terms (divide the numerator and denominator by the GCF)
a/10ⁿ = (a ÷ g)/(10ⁿ ÷ g)
\dfrac{a}{10^{n}} = \dfrac{a \div g}{10^{n} \div g}, \quad g = \gcd(a,\ 10^{n})
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac>
<mi>a</mi>
<msup><mn>10</mn><mi>n</mi></msup>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>÷</mo><mi>g</mi></mrow>
<mrow><msup><mn>10</mn><mi>n</mi></msup><mo>÷</mo><mi>g</mi></mrow>
</mfrac>
</mrow>
</math>
a/10^n = (a -: g)/(10^n -: g)
(a/g)/(10^n/g)
(a/g)/(10^n/g);
N = a/g; D = 10^n/g;
a/10^n = (a÷g)/(10^n÷g)
Changing an improper fraction to a mixed number (for decimals greater than 1)
N = q × D + r
N = q \times D + r
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>N</mi>
<mo>=</mo>
<mi>q</mi>
<mo>×</mo>
<mi>D</mi>
<mo>+</mo>
<mi>r</mi>
</mrow>
</math>
N = q xx D + r
q*d + r (* D (the derivative operator) is reserved in Mathematica, so a lowercase d is used *)
N := q*d + r; # D (the derivative operator) is reserved in Maple, so a lowercase d is used
N = q*D + r;
N = q × D + r
How to have ChatGPT do the calculation
You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Convert the decimal 1.375 to a fraction. 1. The fraction in simplest form 2. The same value as a mixed number (with a whole-number part) 3. The steps (the fraction with a power-of-10 denominator, and the greatest common factor used to simplify it) Use Python's fractions module (pass the value to Fraction as a string), and show the formulas you used and the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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