Fraction Calculator

Cum se utilizează acest calculator

1. Introduceți Numerator 1
2. Introduceți Denominator 1
3. Introduceți Operation
4. Introduceți Numerator 2
5. Introduceți Denominator 2
6. Faceți clic pe butonul Calculați
7. Citiți rezultatul afișat sub calculator

Understanding Fractions: The Basics

A fraction represents a part of a whole. It is written as numerator / denominator, where the numerator is how many parts you have and the denominator is how many equal parts make up the whole.

Types of fractions:

• Proper fraction: Numerator < denominator (e.g., 3/4). Value is less than 1.
• Improper fraction: Numerator ≥ denominator (e.g., 7/4). Value is greater than or equal to 1.
• Mixed number: Whole number + proper fraction (e.g., 1¾). Equivalent to 7/4.
• Equivalent fractions: Different fractions that represent the same value (e.g., 1/2 = 2/4 = 3/6 = 50/100).

Converting between forms:

• Improper to mixed: Divide numerator by denominator. 7 ÷ 4 = 1 remainder 3 → 1¾
• Mixed to improper: (Whole × denominator) + numerator. 1¾ = (1 × 4) + 3 = 7/4
• Fraction to decimal: Divide numerator by denominator. 3/8 = 0.375
• Decimal to fraction: Write decimal over its place value. 0.375 = 375/1000 = 3/8 (after simplification)

How to Add and Subtract Fractions

Adding and subtracting fractions requires a common denominator — both fractions must express parts of the same-sized whole before you can combine them.

Step-by-step: Adding fractions with different denominators

1. Find the Least Common Denominator (LCD) — the smallest number divisible by both denominators
2. Convert each fraction to an equivalent fraction with the LCD
3. Add (or subtract) the numerators; keep the denominator
4. Simplify by dividing numerator and denominator by their Greatest Common Divisor (GCD)

Example: 2/3 + 3/4

• LCD of 3 and 4 = 12
• 2/3 = 8/12 (multiply both by 4); 3/4 = 9/12 (multiply both by 3)
• 8/12 + 9/12 = 17/12
• Simplify: 17 and 12 share no common factors → 17/12 (or 1 5/12 as a mixed number)

Example: 5/6 − 1/4

• LCD of 6 and 4 = 12
• 5/6 = 10/12; 1/4 = 3/12
• 10/12 − 3/12 = 7/12 (already in lowest terms)

Finding the LCD efficiently: If the denominators share no common factors, the LCD = their product (3 × 4 = 12). If they share factors, use the formula: LCD = (a × b) ÷ GCD(a, b). For 6 and 4: GCD = 2, LCD = (6 × 4) ÷ 2 = 12.

Multiplying and Dividing Fractions

Multiplication and division of fractions are actually simpler than addition — they don't require a common denominator.

Multiplication: Multiply numerators together, multiply denominators together.

Formula: (a/b) × (c/d) = (a × c) / (b × d)

Example: 3/5 × 2/7 = (3 × 2) / (5 × 7) = 6/35

Example: 4/9 × 3/8

• Naively: (4 × 3) / (9 × 8) = 12/72
• Better: Cancel common factors first (cross-cancel): 4 and 8 share factor 4; 3 and 9 share factor 3. Simplify: (1/3) × (1/2) = 1/6
• Cross-canceling first avoids working with large numbers

Division: Multiply by the reciprocal of the divisor. "Keep, Change, Flip."

Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)

Example: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1¼

Why does "flip and multiply" work? Division by a fraction is multiplication by its reciprocal, because the reciprocal × the fraction = 1. Dividing by (2/3) is the same as multiplying by (3/2). This identity makes fraction division as simple as multiplication.

Simplifying Fractions: Finding the GCD

A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factors other than 1. Simplifying means dividing both by their Greatest Common Divisor (GCD).

Example: Simplify 48/72

Method 1 — Factor both: 48 = 2⁴ × 3; 72 = 2³ × 3². GCD = 2³ × 3 = 24. 48/24 = 2; 72/24 = 3. Simplified: 2/3.

Method 2 — Euclidean algorithm (most efficient for large numbers):

1. Divide the larger number by the smaller: 72 ÷ 48 = 1, remainder 24
2. Replace larger with smaller, smaller with remainder: GCD(48, 24)
3. Divide: 48 ÷ 24 = 2, remainder 0
4. When remainder = 0, the GCD is the last divisor: GCD = 24

The Euclidean algorithm is one of the oldest algorithms in mathematics (from Euclid's Elements, ~300 BC) and is still used in modern computing.

Quick prime check for simplification: If both numbers are even, divide both by 2. If both end in 0 or 5, divide both by 5. If both digits sum to a multiple of 3, divide by 3. Repeat until no common factor remains.

Practical examples — common fractions in lowest terms:

Fractions in Real Life: Practical Applications

Fractions appear constantly in everyday life — cooking, construction, finance, and medicine all rely on accurate fraction understanding.

Cooking and recipes: Scaling a recipe up or down requires fraction multiplication. A recipe calls for 2/3 cup of flour and you want to make 1.5× the recipe: 2/3 × 3/2 = 6/6 = 1 cup. Halving: 2/3 × 1/2 = 2/6 = 1/3 cup.

Construction and measurement: Lumber and hardware in the US are measured in fractional inches. Adding boards: 3⅝" + 4¾" = 3 5/8 + 4 6/8 = 7 11/8 = 8 3/8 inches. Subtracting clearance: a 2¼" pipe in a 2½" gap leaves 2/4" = 1/4" clearance.

Finance: Fractional shares are now standard in investment accounts. Interest rates are expressed in fractions: a mortgage rate of 6⅜% = 6.375%. Federal Reserve rate decisions use fractions of a percentage point (25 basis points = 1/4 percent).

Medical dosing: Medication is frequently dosed in fractions of milligrams or milliliters. A dose of 1/4 tablet, or 0.5 mg/kg body weight, requires fraction arithmetic for safety-critical calculations.

Probability: Probability is inherently fractional. If 5 out of 12 widgets are defective, the probability of choosing a defective one is 5/12 ≈ 0.417 or 41.7%. The sum of all probabilities = 1 (the whole), making fraction arithmetic fundamental to statistics.

Frequently Asked Questions