Simplify Fractions Calculator (Reduce to Lowest Terms)

Enter the fraction just as it looks (top box = numerator, bottom box = denominator). For a mixed number such as \(2\dfrac{21}{98}\), also use the whole number box to the left of the fraction. For an ordinary fraction, leave the whole number box blank.

For a negative fraction, put the minus sign in the whole number box. Decimals such as 1.5 are fine; the numerator and denominator are multiplied by 10, 100 and so on to make whole numbers before simplifying. The denominator cannot be 0.

Result

Enter the numerator and denominator on the left and press "Calculate". The result will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formula
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the numerator and denominator, and the fraction is simplified to lowest terms (the form that cannot be simplified any further)
Mixed numbers (with a whole number part) such as \(2\dfrac{21}{98}\) can be simplified as they are, and the answer is shown both as an improper fraction (numerator at least as large as the denominator) and as a mixed number
Negative fractions and fractions with decimals, such as \(\dfrac{1.5}{2}\), work too. The value as a decimal and each step of the work are also shown
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

Simplifying does not change the value of a fraction. It divides the numerator and denominator by the same number (their greatest common factor) so that the fraction looks as simple as possible.

What is this calculation used for?

Avoiding lost points for unsimplified answers on tests (math class)

On math tests, fraction answers are normally expected in lowest terms, and forgetting to simplify often costs points.
"When you get an answer, check at the end that the numerator and denominator have no common factor left." This habit is a basic move from elementary school fractions all the way to college entrance exams. Dividing by the greatest common factor finishes it in one step, so checking is quick too.

The "16:9" of screens and TVs comes from simplifying

A Full HD screen is 1920 × 1080 pixels, but its ratio is called 16:9: \(\dfrac{1920}{1080}\) simplified by the greatest common factor 120 is \(\dfrac{16}{9}\).
Writing a ratio in lowest terms lets you see at a glance whether two shapes are the same, so this idea is used every day in video, design and printing.

Reading the gear ratio of a bicycle or machine

On a bicycle with 48 teeth on the front gear and 18 on the rear, the gear ratio is \(\dfrac{48}{18}\) simplified to \(\dfrac{8}{3}\). You can see right away that "one turn of the pedals turns the rear wheel \(\dfrac{8}{3}\) times (about 2.67 turns)".
In designing machines with gears, the simplified ratio is the basic language for describing the specs.

Remembering a recipe as a ratio to make any amount

Frozen orange juice concentrate usually says to mix a 12-ounce can with 36 ounces of water (3 cans). Simplify \(\dfrac{12}{36}\) to \(\dfrac{1}{3}\) and remember it as "1 part concentrate to 3 parts water", and you can make the same taste for 2 people or for 10.
Professional recipes are often written as ratios like "1:3" because they have been simplified to the simplest ratio.

Reading a probability as "once in so many times"

The probability that two dice add up to 3 is \(\dfrac{2}{36}\) if you just count, but simplified to \(\dfrac{1}{18}\), you can read it as "about once in 18 rolls".
Probabilities often come out of a calculation unsimplified, and from lottery odds to insurance planning, simplifying is essential whenever you explain a probability to someone.

Formula

Simplifying (divide the numerator and denominator by the GCF)

Standard notation (the usual math form)

\(N\) \(=\) \(n\) \(\div\) \(g\)

\(D\) \(=\) \(d\) \(\div\) \(g\)

In words (symbols replaced with words)

③ \(N\): simplified numerator \(=\) ① \(n\): original numerator \(\div\) ② \(g\): greatest common factor

⑥ \(D\): simplified denominator \(=\) ④ \(d\): original denominator \(\div\) ⑤ \(g\): greatest common factor

The formula in words

① Divide the \(n\): original numerator

② by the \(g\): greatest common factor

③ to get the \(N\): simplified numerator

④ Divide the \(d\): original denominator

⑤ by the same \(g\): greatest common factor

⑥ to get the \(D\): simplified denominator

Quick example

Simplifying \(\dfrac{12}{18}\) (the greatest common factor of 12 and 18 is 6)

\(N\): simplified numerator \(=\) original numerator (12) \(\div\) GCF (6)

\(D\): simplified denominator \(=\) original denominator (18) \(\div\) GCF (6)

\(12 \div 6 = 2,\quad 18 \div 6 = 3\)

\(\dfrac{12}{18} = \dfrac{2}{3}\)

Key idea

A fraction keeps the same value if you multiply or divide its numerator and denominator by the same number. Simplifying uses this property. If you divide by the greatest common factor, one division takes you straight to the form that cannot be simplified any further (lowest terms). Dividing by a smaller common factor (for example, 2) is not wrong, but then you have to simplify again.

How to find the greatest common factor \(g\)

Standard notation (the usual math form)

\(g\) \(=\) \(\gcd(\) \(n\) \(,\) \(d\) \()\)

In words (symbols replaced with words)

③ \(g\): greatest common factor \(=\) \(\gcd(\) ① \(n\): numerator \(,\) ② \(d\): denominator \()\)

The formula in words

① Of the numbers that divide both the \(n\): numerator

② and the \(d\): denominator evenly (the common factors), the largest one is the

③ \(g\): greatest common factor

Quick example

The greatest common factor of 12 and 18 (factor each into primes and multiply the prime factors they share)

\(g\): greatest common factor \(=\) \(\gcd(\) numerator (12) \(,\) denominator (18) \()\)

\(12 = 2 \times 2 \times 3,\quad 18 = 2 \times 3 \times 3\)

\(g = 2 \times 3 = 6\)

Key idea

For small numbers, it is enough to list the factors and find the largest one they share (the factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 18 are 1, 2, 3, 6, 9, 18, so the largest shared one is 6). For large numbers, you can use prime factorization and multiply the shared prime factors, or use the Euclidean algorithm (divide the larger number by the smaller one and keep replacing with the remainder). In Excel, the GCD function finds it in one step, and in Python, math.gcd does.

Changing a mixed number to an improper fraction

Standard notation (the usual math form)

\(N\) \(=\) \(w\) \(\times\) \(d\) \(+\) \(n\)

In words (symbols replaced with words)

④ \(N\): numerator of the improper fraction \(=\) ① \(w\): whole number part \(\times\) ② \(d\): denominator \(+\) ③ \(n\): numerator

The formula in words

① Multiply the \(w\): whole number part

② by the \(d\): denominator

③ and add the \(n\): numerator

④ to get the \(N\): numerator of the improper fraction (the denominator stays the same)

Quick example

Changing the mixed number \(2\dfrac{21}{98}\) (two and twenty-one ninety-eighths) to an improper fraction

\(N\): numerator of the improper fraction \(=\) whole number part (2) \(\times\) denominator (98) \(+\) numerator (21)

\(2 \times 98 + 21 = 217\)

\(2\dfrac{21}{98} = \dfrac{217}{98}\)

Key idea

A mixed number is hard to simplify as it is, so first change it to an improper fraction (a fraction whose numerator is at least as large as its denominator). To see why the formula works, think of the whole number part as that many fractions equal to 1 with the same denominator (2 is two of \(\dfrac{98}{98}\), which is \(\dfrac{196}{98}\)).

Changing an improper fraction to a mixed number

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 (the whole number part) \(\times\) ② \(D\): denominator \(+\) ④ \(r\): remainder (the new numerator)

The formula in words

① Divide the \(N\): numerator of the improper fraction

② by the \(D\): denominator

③ The \(q\): quotient becomes the whole number part of the mixed number, and

④ the \(r\): remainder becomes the numerator of the fraction part (the denominator stays \(D\))

Quick example

Changing the improper fraction \(\dfrac{31}{14}\) to a mixed number (31 ÷ 14 = 2 remainder 3)

numerator of the improper fraction (31) \(=\) quotient (2) \(\times\) denominator (14) \(+\) remainder (3)

\(31 = 2 \times 14 + 3\)

\(\dfrac{31}{14} = 2\dfrac{3}{14}\)

Key idea

Using the quotient and remainder of "numerator ÷ denominator", you can change an improper fraction to a mixed number. For a negative fraction, drop the sign, do the same calculation, and put the sign back in front of the whole mixed number at the end (for example, \(-\dfrac{31}{14}\) → \(-2\dfrac{3}{14}\)).

To simplify a fraction, just divide the numerator and denominator by their greatest common factor. Change a mixed number to an improper fraction before simplifying, then turn the result back into a mixed number or a decimal if you need to.

Symbols and terms

Symbols

\(n\)	lowercase n	The numerator (top number) of the fraction before simplifying. (Example - the 21 in \(\dfrac{21}{98}\))
\(d\)	lowercase d	The denominator (bottom number) of the fraction before simplifying. (Example - the 98 in \(\dfrac{21}{98}\))
\(g\)	g	The greatest common factor of the numerator and denominator. To simplify, divide both by this number. (Example - for 21 and 98, it is 7)
\(N\)	capital N	The new numerator after a calculation. On this page it is the simplified numerator, or the numerator you get when changing a mixed number to an improper fraction.
\(D\)	capital D	The new denominator after simplifying. (Example - the 14 in \(\dfrac{3}{14}\), the simplified form of \(\dfrac{21}{98}\))
\(w\)	w	The whole number part of a mixed number, from the first letter of "whole". (Example - the 2 in \(2\dfrac{3}{14}\))
\(q\)	q	The quotient when the numerator is divided by the denominator, from the first letter of "quotient". It becomes the whole number part of the mixed number.
\(r\)	r	The remainder when the numerator is divided by the denominator, from the first letter of "remainder". It becomes the numerator of the fraction part of the mixed number.
\(\gcd(n, d)\)	G-C-D of n and d	The notation for "the greatest common factor of \(n\) and \(d\)". GCD stands for greatest common divisor, another name for the greatest common factor.

Terms

simplifying	Dividing the numerator and denominator of a fraction by a common factor to make the fraction simpler. Also called reducing a fraction. The value itself does not change.
lowest terms	A fraction is in lowest terms (or simplest form) when it cannot be simplified any further. The greatest common factor of its numerator and denominator is 1. When a test says "simplify your answer", this is the form to use.
common factor	A factor shared by two or more numbers. (Example - the common factors of 12 and 18 are 1, 2, 3 and 6)
greatest common factor (GCF)	The largest of the common factors. It is also called the greatest common divisor (GCD).
improper fraction	A fraction such as \(\dfrac{31}{14}\) whose numerator is equal to or larger than its denominator. Its value is 1 or more.
mixed number	A number such as \(2\dfrac{3}{14}\) written as a whole number part followed by a fraction part. It has the same value as an improper fraction but makes the size easier to picture.
proper fraction	A fraction such as \(\dfrac{3}{14}\) whose numerator is smaller than its denominator. Its value is less than 1. The fraction part of a mixed number is always a proper fraction.
prime factorization	Writing a number as a product of prime numbers (for example, \(12 = 2 \times 2 \times 3\)). Multiplying the prime factors two numbers share gives their greatest common factor.
Euclidean algorithm	A way to find the greatest common factor by dividing the larger number by the smaller one, then repeating with the pair "divisor and remainder". It is fast even for large numbers.

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.

What a fraction is (Grades 3–4)	Knowing that a fraction stands for "some number of the equal parts that 1 is split into" Knowing the names and jobs of the numerator (top number) and the denominator (bottom number)
Factors, common factors and the GCF (Grades 4–6)	Being able to list all the factors of a number (for example, the factors of 12 are 1, 2, 3, 4, 6, 12) Knowing that the largest of the factors two numbers share (their common factors) is the greatest common factor
Equivalent fractions, simplifying, improper fractions and mixed numbers (Grades 4–5)	Knowing that multiplying or dividing the numerator and denominator by the same number does not change the value of a fraction Being able to change improper fractions to mixed numbers and back (for example, \(\dfrac{31}{14} = 2\dfrac{3}{14}\))
Division with remainders (Grades 3–4)	Being able to divide with a quotient and a remainder, as in "31 ÷ 14 = 2 remainder 3"

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 simplify (divide the numerator and denominator by the GCF)

Original numerator n	21
Original denominator d	98
Greatest common factor g	=GCD(B1,B2)
Simplified numerator N	=B1/B3
Simplified denominator D	=B2/B3

Table to find the greatest common factor

First number (numerator)	12
Second number (denominator)	18
Greatest common factor g	=GCD(B1,B2)

Table to find the numerator when changing a mixed number to an improper fraction

Whole number part w	2
Numerator n	21
Denominator d	98
Numerator of the improper fraction N	=B1*B3+B2

Table to change an improper fraction to a mixed number

Numerator of the improper fraction N	31
Denominator D	14
Whole number part (quotient) q	=QUOTIENT(B1,B2)
Remainder (numerator of the fraction part) r	=MOD(B1,B2)

After pasting, the upper rows are your inputs and the cells starting with "=" are calculated automatically.
GCD finds the greatest common factor, QUOTIENT finds the quotient (the whole number part), and MOD finds the remainder.
In the first table, for example, B3 shows the greatest common factor 7, B4 the simplified numerator 3, and B5 the simplified denominator 14 (\(\dfrac{21}{98}\) → \(\dfrac{3}{14}\)).
To change a negative fraction to a mixed number, calculate without the 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 simplify (divide the numerator and denominator by the GCF)

Original numerator n	21
Original denominator d	98
Greatest common factor g	=GCD(B1,B2)
Simplified numerator N	=B1/B3
Simplified denominator D	=B2/B3

Table to find the greatest common factor

First number (numerator)	12
Second number (denominator)	18
Greatest common factor g	=GCD(B1,B2)

Table to find the numerator when changing a mixed number to an improper fraction

Whole number part w	2
Numerator n	21
Denominator d	98
Numerator of the improper fraction N	=B1*B3+B2

Table to change an improper fraction to a mixed number

Numerator of the improper fraction N	31
Denominator D	14
Whole number part (quotient) q	=QUOTIENT(B1,B2)
Remainder (numerator of the fraction part) r	=MOD(B1,B2)

The same formulas as in Excel work as is (GCD, QUOTIENT and MOD are all available in Google Sheets too).
Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own fraction.

How to calculate it in Python

from fractions import Fraction
from math import gcd

whole = 2          # whole number part of the mixed number (0 for an ordinary fraction)
numerator = 21     # numerator
denominator = 98   # denominator

improper_numerator = whole * denominator + numerator    # numerator after changing to an improper fraction
g = gcd(improper_numerator, denominator)                # greatest common factor of numerator and denominator
simplified = Fraction(improper_numerator, denominator)  # fraction in lowest terms (Fraction simplifies automatically)
q, r = divmod(abs(simplified.numerator), simplified.denominator)  # whole number part and remainder for the mixed number

print(f"As an improper fraction: {improper_numerator}/{denominator}")
print(f"Greatest common factor: {g}")
print(f"In lowest terms: {simplified}")
print(f"As a mixed number: {q} {r}/{simplified.denominator}")
print(f"As a decimal: {float(simplified)}")

Runs with the standard library only. Fraction is a type that simplifies fractions automatically, and math.gcd is a function that finds the greatest common factor. Change the three numbers at the top and run it.

How to write it in LaTeX and other math languages (copy and paste)

Simplifying (divide the numerator and denominator by the GCF)

n/d = (n ÷ g)/(d ÷ g)

\dfrac{n}{d} = \dfrac{n \div g}{d \div g}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mfrac><mi>n</mi><mi>d</mi></mfrac>
    <mo>=</mo>
    <mfrac>
      <mrow><mi>n</mi><mo>&#x00F7;</mo><mi>g</mi></mrow>
      <mrow><mi>d</mi><mo>&#x00F7;</mo><mi>g</mi></mrow>
    </mfrac>
  </mrow>
</math>

n/d = (n -: g)/(d -: g)

(n/g)/(d/g)

(n/g)/(d/g);

N = n/g; D = d/g;

n/d = (n/g)/(d/g)

How to find the greatest common factor \(g\)

g = gcd(n, d)

g = \gcd(n,\ d)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>g</mi>
    <mo>=</mo>
    <mi>gcd</mi>
    <mo stretchy="false">(</mo>
    <mi>n</mi>
    <mo>,</mo>
    <mi>d</mi>
    <mo stretchy="false">)</mo>
  </mrow>
</math>

g = gcd(n, d)

GCD[n, d]

g := gcd(n, d);

g = gcd(n, d);

g = gcd(n, d)

Changing a mixed number to an improper fraction

N = w × d + n

N = w \times d + n

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>N</mi>
    <mo>=</mo>
    <mi>w</mi>
    <mo>&#x00D7;</mo>
    <mi>d</mi>
    <mo>+</mo>
    <mi>n</mi>
  </mrow>
</math>

N = w xx d + n

w*d + n

N := w*d + n;

N = w*d + n;

N = w × d + n

Changing an improper fraction to a mixed number

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>&#x00D7;</mo>
    <mi>D</mi>
    <mo>+</mo>
    <mi>r</mi>
  </mrow>
</math>

N = q xx D + r

q*d + r  (* lowercase d because D (the derivative operator) is reserved in Mathematica *)

N := q*d + r;  # lowercase d because D (the derivative operator) is reserved in Maple

N = q*D + r;

N = q × D + r

How to have ChatGPT do the calculation

You are a fraction 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).

Simplify the mixed number "2 and 21/98" to its simplest form.
Find each of the following:
1. The mixed number written as an improper fraction
2. The greatest common factor of the numerator and denominator
3. The fraction in lowest terms (both as an improper fraction and as a mixed number)
4. The value as a decimal

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula