General calculator
Fraction Calculator
Free online fraction calculator. Easily add, subtract, multiply, and divide proper, improper, and mixed fractions with step-by-step simplification.
Introduction
Fractions are a fundamental concept in mathematics representing parts of a whole. E.g. 1/2, 3/4. Our Fraction Calculator is a comprehensive tool designed to help you add, subtract, multiply, and divide proper, improper, and mixed fractions. Whether you are checking homework, adjusting a baking recipe, or working on construction dimensions, this calculator simplifies fractions instantly and displays the decimal equivalent alongside mixed number forms.
How to Use
To use the Fraction Calculator, select your preferred operation (+, -, *, or /) from the drop-down menu. Enter the numerator (top number) and denominator (bottom number) for Fraction 1 and Fraction 2. Denominators cannot be zero. Click "Calculate" to view the resulting fraction, its simplified form, mixed number equivalent, and decimal representation.
Formula
The mathematical formulas for operating on two fractions (a/b and c/d) are: Addition: (ad + bc) / bd; Subtraction: (ad - bc) / bd; Multiplication: ac / bd; Division: ad / bc (by multiplying the reciprocal of the divisor). After performing the operation, the greatest common divisor (GCD) of the resulting numerator and denominator is calculated to simplify the fraction to its lowest terms.
Examples
Example 1 (Addition): 1/3 + 2/5. Using the formula (1×5 + 2×3) / (3×5) = (5 + 6) / 15 = 11/15. This is in simplest form. Example 2 (Division): 3/4 ÷ 2/3. Multiply by reciprocal: 3/4 × 3/2 = 9/8. Simplified as mixed number: 1 1/8, or 1.125 as a decimal.
Results Explained
The output displays: (1) Simplified Fraction: the result simplified to its lowest terms. (2) Mixed Number: the result expressed as a whole number plus a proper fraction. (3) Decimal: the decimal equivalent of the fraction, rounded to four decimal places.
Related Calculators
Fractions Explained: Proper, Improper, and Mixed
A fraction consists of two parts: a numerator (the top number indicating how many parts you have) and a denominator (the bottom number indicating how many equal parts divide the whole). Depending on the relationship between these two numbers, fractions are categorized into three main types:
- Proper Fractions: The numerator is smaller than the denominator (e.g., 2/3, 5/8). These fractions represent values strictly between 0 and 1.
- Improper Fractions: The numerator is greater than or equal to the denominator (e.g., 7/4, 9/5, 4/4). These fractions represent values equal to or greater than 1.
- Mixed Numbers: A combination of a whole number and a proper fraction (e.g., 1 3/4, 4 1/2). Mixed numbers can always be converted into improper fractions and vice versa.
Adding and Subtracting Fractions
Adding and subtracting fractions requires one key step: ensuring the denominators are the same. If the denominators are already identical (like denominators), you simply add or subtract the numerators and keep the denominator constant:
Example: 2/7 + 3/7 = (2 + 3)/7 = 5/7
If the denominators are different (unlike denominators), you must find a common denominator. The easiest way to find a common denominator is to multiply the two denominators together (or find the Least Common Multiple, LCM). You then adjust the numerators proportionally:
Formula: a/b ± c/d = (ad ± bc) / bd
Multiplying Fractions
Unlike addition and subtraction, multiplying fractions is straightforward and does not require a common denominator. You simply multiply the numerators together and the denominators together:
Formula: (a/b) × (c/d) = (a × c) / (b × d)
Once multiplied, you should simplify the resulting fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor.
Dividing Fractions
Dividing fractions utilizes the concept of a reciprocal. To divide by a fraction, you multiply by its reciprocal (flip the second fraction upside down). This technique is commonly summarized by the mnemonic "Keep, Change, Flip":
- Keep the first fraction as it is.
- Change the division sign to a multiplication sign.
- Flip the second fraction (find its reciprocal).
Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)
Four Detailed Worked Examples
Example 1: Adding Fractions (Unlike Denominators)
Let's add 2/3 and 1/4:
1. Find common denominator: 3 × 4 = 12.
2. Convert 2/3: Multiply numerator and denominator by 4 → 8/12.
3. Convert 1/4: Multiply numerator and denominator by 3 → 3/12.
4. Add: 8/12 + 3/12 = 11/12.
The result is 11/12, which cannot be simplified further.
Example 2: Subtracting Fractions with Mixed Numbers
Let's subtract 1 1/2 from 3 1/4:
1. Convert mixed numbers to improper fractions:
3 1/4 = (3 × 4 + 1)/4 = 13/4.
1 1/2 = (1 × 2 + 1)/2 = 3/2.
2. Find common denominator: 4.
3. Convert 3/2 to have denominator 4 → 6/4.
4. Subtract: 13/4 - 6/4 = 7/4.
5. Convert back to mixed number: 7 ÷ 4 = 1 with a remainder of 3 → 1 3/4.
Example 3: Multiplying Fractions and Simplifying
Let's multiply 4/5 by 5/8:
1. Multiply numerators: 4 × 5 = 20.
2. Multiply denominators: 5 × 8 = 40.
3. Resulting fraction is 20/40.
4. Simplify: The Greatest Common Divisor (GCD) of 20 and 40 is 20. Divide numerator and denominator by 20 → 1/2.
Example 4: Dividing Fractions
Let's divide 5/6 by 2/3:
1. Keep 5/6, change division to multiplication, flip 2/3 to 3/2.
2. Multiply: (5/6) × (3/2) = 15/12.
3. Simplify: The GCD of 15 and 12 is 3. Divide by 3 → 5/4.
4. Convert to mixed number: 1 1/4 (or 1.25 as a decimal).
Tips for Working with Fractions
- Always simplify: Always check if the numerator and denominator share any common factors. E.g. both are even, divide by 2.
- Check for mixed number inputs: If you are working with a mixed number like 2 3/5, convert it to an improper fraction first: (2 × 5 + 3)/5 = 13/5, then carry out the operations.
- Do not add denominators: A very common mistake is adding numerators and adding denominators (e.g. 1/2 + 1/2 = 2/4 = 1/2, which is wrong!). Keep denominators same and add numerators.
Frequently Asked Questions
What is a fraction?
A fraction represents a part of a whole or, more generally, any number of equal parts. It consists of a numerator on top and a non-zero denominator on the bottom.
What is a proper fraction?
A proper fraction is a fraction where the numerator (top number) is strictly smaller than the denominator (bottom number), representing a value less than 1.
What is an improper fraction?
An improper fraction is a fraction where the numerator is greater than or equal to the denominator, representing a value greater than or equal to 1.
What is a mixed number?
A mixed number is an expression consisting of a whole number and a proper fraction, representing their sum (e.g. 2 1/2 represents 2 + 1/2).
How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, 3 1/2 becomes (3 × 2 + 1) / 2 = 7/2.
How do you convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the numerator of the proper fraction, and the denominator remains the same.
Why can the denominator of a fraction never be zero?
The denominator represents the number of equal parts a whole is divided into. Dividing anything into zero parts is mathematically undefined and impossible.
What is the greatest common divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is used to simplify fractions to their simplest terms.
How do you simplify a fraction?
To simplify a fraction, find the GCD of the numerator and the denominator, and then divide both the numerator and the denominator by that number.
What is a common denominator?
A common denominator is a shared multiple of the denominators of two or more fractions, which is required to add or subtract them.
Can a fraction represent a negative number?
Yes, if either the numerator or the denominator (but not both) is negative, the entire fraction represents a negative value.
How do you multiply two fractions?
Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. No common denominator is required.
How do you divide two fractions?
Multiply the first fraction by the reciprocal (flipped version) of the second fraction. Keep, Change, Flip.
Is a decimal a fraction?
Yes, any decimal can be written as a fraction. For example, 0.75 can be written as 75/100, which simplifies to 3/4.
What is the reciprocal of a fraction?
The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 3/4 is 4/3.