Fraction Calculator: Add, Subtract, Multiply, and Divide Fractions

Table of Contents - Fraction

How to Use This Calculator
The Core Principle: Fraction Arithmetic
How to Calculate with Fractions Manually
Real-World Applications
Scenarios People Actually Run Into
Trade-Offs and Decisions People Underestimate
Common Mistakes and How to Recover
Related Topics
How This Calculator Works
FAQs

How to Use This Calculator - Fraction

Enter fractions using the numerator and denominator fields. The calculator supports multiple fractions (up to 5) for chain calculations.

For each fraction, enter the Numerator (top number) and Denominator (bottom number).

Select the Operation between fractions: + (add), - (subtract), × (multiply), or ÷ (divide).

Use "Add Fraction" to include additional fractions in your calculation. Use the remove button to delete fractions (minimum 2 required).

Click "Calculate" to see results. The output displays:

The complete expression (e.g., "3/4 + 1/2 = 5/4")
The answer in simplified form
The decimal equivalent
Step-by-step solution showing how the answer was derived

A "Clear" button resets all fields.

The Core Principle: Fraction Arithmetic

Fractions represent parts of a whole. The numerator (top) tells how many parts you have; the denominator (bottom) tells how many parts make a whole.

Addition and subtraction require common denominators because you can only add like parts. You can't directly add 1/3 and 1/4 because thirds and fourths are different sizes.

Multiplication is straightforward: multiply numerators together and denominators together. No common denominator needed because you're finding a fraction of a fraction.

Division by a fraction means multiplying by its reciprocal. Dividing by 1/2 is the same as multiplying by 2/1.

Simplification uses the greatest common divisor (GCD) to reduce fractions to lowest terms. 4/8 simplifies to 1/2 because both 4 and 8 are divisible by 4.

How to Calculate with Fractions Manually

Addition (finding common denominator): 3/4 + 1/6

Step 1: Find LCD of 4 and 6 = 12 Step 2: Convert fractions: 3/4 = 9/12, 1/6 = 2/12 Step 3: Add numerators: 9/12 + 2/12 = 11/12

Subtraction: 5/8 - 1/3

Step 1: Find LCD of 8 and 3 = 24 Step 2: Convert: 5/8 = 15/24, 1/3 = 8/24 Step 3: Subtract: 15/24 - 8/24 = 7/24

Multiplication: 2/3 × 4/5

Step 1: Multiply numerators: 2 × 4 = 8 Step 2: Multiply denominators: 3 × 5 = 15 Step 3: Result: 8/15 (already simplified)

Division: 3/4 ÷ 2/5

Step 1: Flip the second fraction: 2/5 → 5/2 Step 2: Multiply: 3/4 × 5/2 = 15/8 Step 3: Convert if needed: 15/8 = 1 7/8

Simplification: Find GCD of numerator and denominator, divide both.

12/18: GCD(12, 18) = 6 12 ÷ 6 = 2, 18 ÷ 6 = 3 Simplified: 2/3

Real-World Applications

Cooking and recipes. A recipe calls for 2/3 cup of flour but you're doubling it. 2/3 × 2 = 4/3 = 1 1/3 cups. Or halving: 2/3 × 1/2 = 2/6 = 1/3 cup.

Construction and woodworking. A board is 5 3/4 inches wide. You need to remove 1/2 inch from each side. 5 3/4 - 1/2 - 1/2 = 5 3/4 - 1 = 4 3/4 inches remaining.

Financial calculations. You own 3/8 of a property. Your sibling owns 2/8 (1/4). Together: 3/8 + 2/8 = 5/8. The remaining 3/8 belongs to your parents.

Time calculations. You worked 2 1/2 hours Monday and 3 1/4 hours Tuesday. Total: 2 1/2 + 3 1/4 = 5/2 + 13/4 = 10/4 + 13/4 = 23/4 = 5 3/4 hours.

Probability. The chance of event A is 1/4. The chance of event B is 1/3. If independent, probability of both: 1/4 × 1/3 = 1/12.

Scenarios People Actually Run Into

The pizza problem. You have 3/4 of a pizza. Your friend takes 1/3 of what you have. How much did they take? 3/4 × 1/3 = 3/12 = 1/4 of the whole pizza. You have 3/4 - 1/4 = 2/4 = 1/2 left.

The measurement dilemma. Your tape measure shows 7/16 inch, but your plans say 3/8 inch. Which is larger? Convert: 3/8 = 6/16. So 7/16 > 6/16. The piece is 1/16 inch too big.

The recipe scaling nightmare. A recipe serves 8 but you need to serve 6. Multiply all ingredients by 6/8 = 3/4. The 2/3 cup of sugar becomes 2/3 × 3/4 = 6/12 = 1/2 cup.

The stock split confusion. You owned 50 shares. After a 3-for-2 split, you have 50 × 3/2 = 150/2 = 75 shares. Each share is worth 2/3 of the original value.

The medication dosage. Adult dose is 1 tablet. Child dose is 1/2 tablet. For a smaller child, 3/4 of the child dose: 1/2 × 3/4 = 3/8 tablet.

Trade-Offs and Decisions People Underestimate

Fractions versus decimals. 1/3 is exact; 0.333... is approximate. For calculations requiring precision (especially with thirds, sixths, sevenths), fractions preserve accuracy.

Simplifying early versus late. Cross-canceling during multiplication simplifies work. 4/9 × 3/8: cancel 3 and 9 to get 4/3 × 1/8 = 4/24. Or: 4/9 × 3/8 = 12/72, then simplify to 1/6. Same answer, different paths.

Mixed numbers versus improper fractions. Mixed numbers (2 1/2) are easier to visualize. Improper fractions (5/2) are easier to calculate with. Convert as needed.

Mental math shortcuts. For common fractions, memorize equivalents: 1/4 = 0.25, 1/3 ≈ 0.33, 3/8 = 0.375. This speeds estimation.

When to use a calculator. Simple fractions with small denominators are easy manually. Complex fractions with large, unfamiliar denominators benefit from calculator assistance.

Common Mistakes and How to Recover

Adding denominators. 1/4 + 1/3 ≠ 2/7. You must find a common denominator: 3/12 + 4/12 = 7/12.

Forgetting to simplify. 6/8 is correct but incomplete. Always reduce: 6/8 = 3/4.

Flipping the wrong fraction when dividing. In a ÷ b, flip b, not a. 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2.

Mixed number arithmetic errors. Convert to improper fractions first. 2 1/3 + 1 1/2: convert to 7/3 + 3/2 = 14/6 + 9/6 = 23/6 = 3 5/6.

LCD mistakes. The least common denominator of 6 and 8 is 24, not 48. Using larger common denominators works but creates bigger numbers to simplify later.

How This Calculator Works

GCD calculation (Euclidean algorithm):

gcd(a, b):
  while b ≠ 0:
    temp = b
    b = a mod b
    a = temp
  return a

Reduction:

reduced = numerator/gcd, denominator/gcd

Addition:

n1/d1 + n2/d2 = (n1×d2 + n2×d1) / (d1×d2)
Then reduce

Subtraction:

n1/d1 - n2/d2 = (n1×d2 - n2×d1) / (d1×d2)
Then reduce

Multiplication:

n1/d1 × n2/d2 = (n1×n2) / (d1×d2)
Then reduce

Division:

n1/d1 ÷ n2/d2 = (n1×d2) / (d1×n2)
Then reduce

For chain calculations, operations are performed left to right, accumulating the result.

All calculations happen locally in your browser.

FAQs

Why do I need a common denominator for addition but not multiplication?

Addition combines like quantities. You can only add thirds to thirds. Multiplication finds a portion of a portion—3/4 of 1/2 means a different operation entirely.

How do I convert a decimal to a fraction?

0.75 = 75/100 = 3/4. For repeating decimals: 0.333... = 1/3. Some decimals (like π) cannot be expressed as fractions.

What's an improper fraction versus a mixed number?

Improper fraction: numerator ≥ denominator (7/4). Mixed number: whole + fraction (1 3/4). They represent the same value.

How do I compare fractions?

Find a common denominator, then compare numerators. Or convert to decimals. 3/8 vs 2/5: 15/40 vs 16/40, so 2/5 > 3/8.

What if my answer has a larger denominator than the inputs?

That's normal for addition and subtraction before simplification. 1/4 + 1/6 = 6/24 + 4/24 = 10/24, which simplifies to 5/12.

Can fractions have negative numbers?

Yes. -3/4 and 3/-4 both equal -0.75. Convention places the negative sign with the numerator or in front of the fraction.

What's a unit fraction?

A fraction with numerator 1, like 1/2, 1/3, 1/4. Ancient Egyptians used only unit fractions, expressing other fractions as sums.

How do I handle zero in fractions?

0/n = 0 for any n. n/0 is undefined (division by zero is not allowed).

What are equivalent fractions?

Fractions that represent the same value: 1/2 = 2/4 = 3/6 = 50/100. Multiply or divide both numerator and denominator by the same number to create equivalent fractions.

How do I add mixed numbers?

Convert to improper fractions first, add, then convert back if desired. 2 1/3 + 1 1/2 = 7/3 + 3/2 = 14/6 + 9/6 = 23/6 = 3 5/6.

What's the difference between proper and improper fractions?

Proper fractions have numerator less than denominator (3/4, 2/5). Improper fractions have numerator greater than or equal to denominator (5/3, 7/4). Both are valid representations.

When should I use fractions versus decimals?

Fractions preserve exactness (1/3 stays exact; 0.333... is approximate). Decimals are easier for some calculations and comparisons. Use fractions when precision matters; decimals for quick estimation.

How do I convert between fractions and percentages?

Multiply the fraction by 100 to get percentage: 3/4 × 100 = 75%. Divide percentage by 100 to get decimal, then convert: 75% = 0.75 = 3/4.

What are complex fractions and how do I simplify them?

Complex fractions have fractions in the numerator, denominator, or both. Simplify by multiplying numerator and denominator by the LCD of all internal fractions, or by rewriting as division: (1/2)/(3/4) = 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3.

How do I work with ratios expressed as fractions?

Ratios like 3:4 can be written as 3/4. To scale, multiply both parts: 3:4 scaled by 5 = 15:20. To compare ratios, convert to fractions and cross-multiply.

What is a reciprocal and when do I use it?

The reciprocal of a/b is b/a. Reciprocals are used in division (dividing by a fraction = multiplying by its reciprocal) and solving equations where a variable is in the denominator.

How do I compare fractions with different denominators quickly?

Cross-multiply: to compare a/b and c/d, compare a×d with b×c. If a×d > b×c, then a/b > c/d. This works because you're effectively converting to a common denominator without calculating it.

What are the most common fraction-decimal equivalents to memorize?

Key conversions: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/3 ≈ 0.333, 2/3 ≈ 0.667, 1/5 = 0.2, 1/8 = 0.125. These speed up estimation and mental math significantly.

Fraction Calculator — Add, Subtract, Multiply & Divide Fractions

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