Fraction Operations Tool
Our fraction calculator helps you add fractions, subtract fractions, multiply fractions, and divide fractions with ease. Simply enter your fractions, select an operation, and get instant results with step-by-step solutions. Perfect for checking homework, learning to simplify fractions, or converting to decimals.
Notes:
- Denominators must be non-zero positive integers
- Results are automatically simplified to lowest terms
- For division, the second fraction is inverted (reciprocal)
- Mixed numbers can be entered as improper fractions (e.g., 1 1/2 would be entered as 3/2)
Understanding Fractions and How to Calculate with Them
What Are Fractions?
A fraction represents a part of a whole. It consists of two numbers: a numerator (top number) and a denominator (bottom number). The numerator represents how many parts we have, while the denominator shows how many equal parts make up the whole.
For example, in the fraction 3/4:
- 3 is the numerator - it tells us we have 3 parts
- 4 is the denominator - it tells us that the whole is divided into 4 equal parts
This means we have 3 out of 4 equal parts, or three-quarters of the whole.
Types of Fractions
Proper Fractions
The numerator is less than the denominator.
Examples: 1/2, 3/4, 5/8
These fractions represent less than one whole unit.
Improper Fractions
The numerator is greater than or equal to the denominator.
Examples: 5/3, 7/4, 11/8
These fractions represent one or more whole units.
Mixed Numbers
A whole number and a proper fraction combined.
Examples: 1 1/2, 2 3/4, 4 1/3
These represent whole units plus a fractional part.
Other important types of fractions include:
- Equivalent fractions: Different fractions that represent the same value (e.g., 1/2 = 2/4 = 3/6)
- Like fractions: Fractions with the same denominator (e.g., 3/5 and 2/5)
- Unlike fractions: Fractions with different denominators (e.g., 1/2 and 1/3)
- Unit fractions: Fractions with a numerator of 1 (e.g., 1/2, 1/3, 1/4)
Converting Between Fraction Types
Improper Fraction to Mixed Number
- Divide the numerator by the denominator
- The quotient is the whole number part
- The remainder is the new numerator
- The denominator stays the same
Example: Convert 11/4 to a mixed number
11 ÷ 4 = 2 with remainder 3
So 11/4 = 2 3/4
Mixed Number to Improper Fraction
- Multiply the whole number by the denominator
- Add the result to the numerator
- This is your new numerator
- The denominator stays the same
Example: Convert 2 3/4 to an improper fraction
(2 × 4) + 3 = 11
So 2 3/4 = 11/4
Simplifying Fractions
A fraction is in its simplest form (or lowest terms) when the numerator and denominator have no common factors other than 1. To simplify a fraction:
- Find the greatest common divisor (GCD) of the numerator and denominator
- Divide both the numerator and denominator by this GCD
Example: Simplify 12/18
The GCD of 12 and 18 is 6
12 ÷ 6 = 2
18 ÷ 6 = 3
Therefore, 12/18 = 2/3 in lowest terms
A quick way to find the GCD is using the Euclidean algorithm, but you can also list all factors of both numbers and find the largest common one.
Basic Fraction Operations
Adding Fractions
Like Fractions (Same Denominator)
- Add the numerators
- Keep the same denominator
- Simplify if possible
Example: 3/8 + 2/8
(3 + 2)/8 = 5/8
Unlike Fractions (Different Denominators)
- Find the least common multiple (LCM) of the denominators
- Convert each fraction to an equivalent fraction with the LCM as denominator
- Add the numerators
- Keep the LCM as the denominator
- Simplify if possible
Example: 1/4 + 1/6
LCM of 4 and 6 is 12
1/4 = 3/12, 1/6 = 2/12
3/12 + 2/12 = 5/12
Subtracting Fractions
Like Fractions (Same Denominator)
- Subtract the numerators
- Keep the same denominator
- Simplify if possible
Example: 7/9 - 4/9
(7 - 4)/9 = 3/9 = 1/3
Unlike Fractions (Different Denominators)
- Find the least common multiple (LCM) of the denominators
- Convert each fraction to an equivalent fraction with the LCM as denominator
- Subtract the numerators
- Keep the LCM as the denominator
- Simplify if possible
Example: 5/6 - 1/4
LCM of 6 and 4 is 12
5/6 = 10/12, 1/4 = 3/12
10/12 - 3/12 = 7/12
Multiplying Fractions
- Multiply the numerators to get the new numerator
- Multiply the denominators to get the new denominator
- Simplify if possible
Example: 2/3 × 4/5
(2 × 4)/(3 × 5) = 8/15
Tip: To simplify before multiplying, you can find common factors between numerators and denominators and cancel them.
Dividing Fractions
- Take the reciprocal of the second fraction (flip the numerator and denominator)
- Multiply the first fraction by this reciprocal
- Simplify if possible
Example: 3/4 ÷ 2/5
3/4 × 5/2 = (3 × 5)/(4 × 2) = 15/8 = 1 7/8
Remember: "Keep, Change, Flip" - keep the first fraction, change division to multiplication, and flip the second fraction.
Finding Least Common Multiple (LCM)
The least common multiple (LCM) is essential for adding and subtracting fractions with different denominators. Here are two methods to find the LCM:
Method 1: Listing Multiples
- List the multiples of each number
- Identify the smallest multiple that appears in both lists
Example: Find the LCM of 4 and 6
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, ...
The smallest common multiple is 12
Method 2: Prime Factorization
- Find the prime factorization of each number
- Take each prime factor to its highest power from either factorization
- Multiply these prime factors together
Example: Find the LCM of 12 and 18
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 4 × 9 = 36
Real-World Applications of Fractions
Cooking and Baking
- Measuring ingredients (1/2 cup, 3/4 teaspoon)
- Adjusting recipes (halving or doubling)
- Converting between measurement systems
- Determining cooking time (2/3 of the original time)
Construction and Carpentry
- Taking measurements (7/8 inch, 3/4 foot)
- Reading blueprints and plans
- Mixing concrete (3:1 ratio of aggregate to cement)
- Determining material quantities
Finance and Business
- Calculating interest rates (5/4 of the prime rate)
- Determining discounts (1/3 off)
- Computing taxes (3/20 of income)
- Profit sharing and investments
- Expressing statistics (3/4 of customers prefer...)
Science and Medicine
- Drug dosages (calculated by fraction of body weight)
- Chemical formulations and dilutions
- Genetic inheritance probabilities (3/4 chance)
- Statistical analysis of experiments
- Physical constants and equations
Music
- Note values (quarter notes, half notes)
- Time signatures (3/4 time, 6/8 time)
- Frequency ratios in harmonics
- Tuning instruments
Sports and Games
- Batting averages in baseball (3/4 or .750)
- Win/loss ratios (won 2/3 of games)
- Field position calculations
- Scoring systems (1/2 point awarded)
- Odds and probabilities in games of chance
Common Fraction Errors and Misconceptions
| Common Error | Incorrect Approach | Correct Approach |
|---|---|---|
| Adding denominators | 1/2 + 1/3 = 2/5 | 1/2 + 1/3 = 3/6 + 2/6 = 5/6 |
| Cancelling incorrectly | 16/64 = 1/4 (cancelling the 6) | 16/64 = (16÷16)/(64÷16) = 1/4 |
| Dividing incorrectly | 2/3 ÷ 4/5 = 2/3 × 4/5 = 8/15 | 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 |
| Converting to decimals incorrectly | 3/4 = 0.34 or 0.43 | 3/4 = 3 ÷ 4 = 0.75 |
| Comparing fractions incorrectly | 1/3 > 1/2 (because 3 > 2) | 1/3 < 1/2 (convert to same denominator: 2/6 < 3/6) |