Fields above the solid line represent the numerator, fields below represent the denominator. Use negative numbers for negative fractions.
Enter whole numbers and fractions for each operand. Leave the whole number as 0 if not needed.
Enter a fraction to reduce it to its simplest (lowest) form.
Enter a decimal number to convert it to a fraction.
Enter a fraction to convert it to a decimal number.
Perform arithmetic with very large integers as numerators and denominators. Enter whole integers only (no decimals or commas).
This page provides six fraction calculators covering every common fraction operation:
All calculators accept positive and negative integers. Input the numerator in the top field and the denominator in the bottom field. Leave unused fields as 0.
A fraction is a number that represents a part of a whole. It consists of a numerator and a denominator. The numerator represents the number of equal parts of a whole, while the denominator represents the total number of equal parts that make up that whole. For example, in the fraction 3/8, the numerator is 3 and the denominator is 8. The fraction represents 3 equal parts out of a possible 8.
The shaded region of this figure represents the fraction 3/8 of a whole pie. The three shaded slices represent the numerator, while all eight equal slices represent the denominator.
Unlike with whole numbers, adding fractions requires that the fractions share a common denominator. The equations below show how to add two fractions. It is often easier to work with simplified fractions. As such, fraction solutions are typically expressed in their simplified forms. For example, the fraction 4⁄8 can be simplified to 1⁄2.
The LCD (Least Common Denominator) method is often more efficient. Instead of multiplying both denominators together (which always works but may produce large numbers), find the LCM of the two denominators. For 1/4 + 1/6, the LCD is 12, so: 3/12 + 2/12 = 5/12.
Fraction subtraction is essentially the same as fraction addition. A common denominator is required for the operation to occur. Refer to the addition section as well as the equations below for clarification.
Multiplying fractions is fairly straightforward. Unlike adding and subtracting, it is not necessary to compute a common denominator in order to multiply fractions. Simply multiply the numerators and denominators of each fraction in the problem by one another.
Often when working with fractions, the resulting fraction can be simplified. An easy shortcut is to cross-cancel before multiplying - divide a numerator and a diagonal denominator by their GCF first. This keeps the numbers smaller.
The process for dividing fractions is similar to that for multiplying fractions. In order to divide fractions, the fraction in the numerator is multiplied by the reciprocal of the fraction in the denominator. The reciprocal of a fraction is simply a fraction with the numerator and denominator switched.
An alternative method for finding a common denominator is to use the number obtained by multiplying the two denominators (bd), reducing the result to the lowest terms by factoring out the GCF of the numerator and denominator.
It is often easier to work with simplified fractions. As such, fraction solutions are typically expressed in their simplified forms. For example, the fraction 8⁄12 can be simplified to 2⁄3.
To simplify a fraction, find the greatest common factor (GCF) - also called the greatest common divisor (GCD) - of the numerator and the denominator, then divide both by that number.
One way to find the GCF is to list the prime factors of each number and identify the common ones. For example: 8 = 2 × 2 × 2 and 12 = 2 × 2 × 3. The common prime factors are 2 × 2 = 4, so GCF(8,12) = 4. The Euclidean algorithm (repeated division) is another efficient approach for larger numbers.
Converting from decimals to fractions is straightforward. It does not require finding a common denominator. Simply count the number of decimal places and use that as a power of 10 for the denominator.
Converting a fraction to a decimal is done by dividing the numerator by the denominator. Some fractions produce terminating decimals (e.g., 3/8 = 0.375), while others produce repeating decimals (e.g., 1/3 = 0.333...). A fraction in lowest terms terminates if and only if the denominator has no prime factors other than 2 and 5.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333... | 33.33...% |
| 2/3 | 0.666... | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
Not all fractions behave the same way. Understanding the different types helps in choosing the right calculation approach:
Two operations underlie almost all fraction arithmetic: finding the Greatest Common Factor (GCF) for simplification and finding the Least Common Multiple (LCM) for addition and subtraction.
The GCF of two integers is the largest integer that divides both without a remainder. There are two reliable methods:
Method 1 - Prime Factorization: Write each number as a product of prime factors. The GCF is the product of the primes they share (using the smaller exponent for each).
Method 2 - Euclidean Algorithm: Repeatedly divide the larger number by the smaller, replacing the larger with the remainder, until the remainder is zero. The last nonzero remainder is the GCF. This is the method used internally by this calculator.
The LCM of two integers is the smallest positive integer that is divisible by both. For fractions, the LCM of the denominators is the Least Common Denominator (LCD).
Method 1 - Using GCF: LCM(a, b) = |a × b| ÷ GCF(a, b). This is the most efficient method.
Method 2 - Prime Factorization: The LCM is the product of each prime factor raised to its highest power across both numbers.
Mixed numbers appear constantly in everyday life - recipes (2 1/2 cups), measurements (5 3/4 inches), time (1 1/4 hours). The calculator handles mixed numbers directly, but understanding the underlying steps is valuable.
Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.
Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the original denominator is the fractional part.
Add 2 3/4 + 1 2/3:
Subtract 3 1/4 − 1 3/4:
The "borrowing" happens automatically when you convert to improper fractions first - the improper fractions absorb the whole-number part, eliminating the need for manual borrowing.
When multiplying fractions, you can simplify before multiplying - cancelling a factor from any numerator with any denominator (not just the numerator and denominator of the same fraction). This is called cross-cancellation and keeps numbers smaller.
Comparing fractions with the same denominator is trivial - the larger numerator wins. Comparing unlike fractions requires converting them to a common denominator first.
Which is larger: 3/4 or 5/7?
A quick alternative is the cross-multiplication method: compare a/b vs. c/d by comparing a×d vs. b×c. If a×d > b×c, then a/b > c/d. For 3/4 vs. 5/7: 3×7=21 and 4×5=20 → 21>20 → 3/4 > 5/7. This avoids finding the LCD entirely.
Plotting fractions on a number line builds intuition for their relative size. To place a/b on a number line:
For improper fractions and mixed numbers, extend the number line past 1. The mixed number 2 3/4 sits three-quarters of the way between 2 and 3. Equivalent fractions (1/2, 2/4, 3/6...) all plot to the exact same point on the number line, confirming they represent the same value.
Fractions are not just a classroom exercise - they appear constantly in practical life:
When converting a fraction to a decimal by long division, the result is either a terminating decimal (the division eventually reaches a remainder of zero) or a repeating decimal (the remainder cycles, producing a repeating pattern of digits).
A fraction in its simplest form produces a terminating decimal if and only if the denominator has no prime factors other than 2 and 5. If the denominator (in lowest terms) contains any other prime factor (3, 7, 11, 13, ...), the decimal repeats.
| Fraction | Denominator factors | Decimal | Type |
|---|---|---|---|
| 1/4 | 2² | 0.25 | Terminating |
| 3/8 | 2³ | 0.375 | Terminating |
| 7/20 | 2² × 5 | 0.35 | Terminating |
| 1/3 | 3 | 0.333... | Repeating |
| 1/7 | 7 | 0.142857142857... | Repeating (6-digit cycle) |
| 5/6 | 2 × 3 | 0.8333... | Repeating |
| 1/11 | 11 | 0.090909... | Repeating (2-digit cycle) |
Every repeating decimal can be converted back to an exact fraction. For example, to convert 0.333... to a fraction: let x = 0.333...; then 10x = 3.333...; subtract: 9x = 3; so x = 3/9 = 1/3.
A mixed number combines a whole number with a proper fraction, such as 2 3/4. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator, then keep the same denominator: 2 3/4 = (2 × 4 + 3)/4 = 11/4. To convert back, divide: 11 ÷ 4 = 2 remainder 3, so 2 3/4.
An improper fraction has a numerator greater than or equal to its denominator, such as 7/4 or 9/9. It is mathematically equivalent to the corresponding mixed number or whole number. Some contexts require expressing an answer as a mixed number; others prefer improper fractions. Both forms are accepted in this calculator's input and output.
The LCD is the Least Common Multiple (LCM) of the denominators. To find the LCM, list multiples of each denominator until you find the first one they share. For example, multiples of 4: 4, 8, 12 - multiples of 6: 6, 12. LCM(4,6) = 12. Alternatively, use the formula LCM(a,b) = |a×b| / GCF(a,b), which is faster for larger numbers. The calculator uses this formula internally.
A fraction is in its simplest (or lowest) form when the GCF of the numerator and denominator is 1 - that is, they share no common factors other than 1. For example, 3/4 is in simplest form because GCF(3,4) = 1, but 6/8 is not because GCF(6,8) = 2. Dividing both by 2 gives 3/4, the simplified form. All results from this calculator are automatically expressed in simplest form.
Yes. A negative fraction can be written with the negative sign on the numerator, denominator, or in front of the fraction - they are all equivalent: −3/4 = 3/(−4) = −(3/4). By convention, the simplified form places the negative sign in the numerator or in front of the fraction, not in the denominator. This calculator accepts negative values in both the numerator and denominator fields.
Division by zero is undefined in mathematics. A fraction a/b represents a ÷ b - asking how many times b fits into a. If b = 0, the question has no meaningful answer: no matter how many times you add 0, you cannot reach any nonzero number. Mathematically, the limit of a/b as b → 0 is either +∞ or −∞ (depending on sign), neither of which is a real number. This is why the calculator rejects any denominator of zero.
Cross-multiplication is a technique for comparing fractions or solving equations involving fractions. To compare a/b and c/d: compute a×d and b×c - if a×d > b×c then a/b > c/d. To solve a/b = c/d for an unknown: cross-multiply to get a×d = b×c, then solve the resulting equation. Cross-multiplication is also the basis of the fraction addition formula: a/b + c/d = (ad + bc)/bd - the numerator is derived from cross-multiplying the fractions and adding the results.
The GCF (Greatest Common Factor, also called GCD - Greatest Common Divisor) is the largest number that divides both integers evenly. It is used to simplify fractions. The LCM (Least Common Multiple) is the smallest number that both integers divide into evenly. It is used to find the common denominator when adding or subtracting fractions. For any two positive integers a and b: GCF(a,b) × LCM(a,b) = a × b. This relationship lets you find one quickly if you already know the other.