Convert mixed number to improper fraction

Convert a mixed number to an improper fraction

Mixed number → Improper fraction Calculator

What is calculated?

This tool converts a mixed number into an improper fraction. The whole part is multiplied by the denominator and added to the numerator to form a fraction where the numerator is greater than or equal to the denominator.

Enter mixed number
Example: 2 6/8 = 22/8 = 11/4
Improper fraction
Mixed number will be converted to an improper fraction

Mixed number Info

Properties

Mixed number conversion:

Whole × Denominator + Numerator Improper fraction Automatic simplification

Note: An improper fraction has a numerator greater than or equal to the denominator.

Examples
Simple: 1 1/2 = 3/2
Standard: 2 3/4 = 11/4
Larger: 3 2/5 = 17/5
With simplification: 2 6/8 = 11/4


Formulas & Rules for Mixed Number Conversion

Main formula
\[n\frac{a}{b} = \frac{n \cdot b + a}{b}\] Conversion formula
Step
\[n \cdot b + a = \text{new numerator}\] Compute numerator
Denominator remains
\[\text{denominator} = b\] Keep unchanged
Definition improper
\[\frac{a}{b}: a \geq b\] Numerator ≥ denominator
Example
\[2\frac{3}{4} = \frac{2 \cdot 4 + 3}{4} = \frac{11}{4}\] Concrete computation
Negative numbers
\[-n\frac{a}{b} = -\frac{n \cdot b + a}{b}\] Sign preserved
Simplification
\[\frac{n \cdot b + a}{b} = \frac{\text{reduced}}{\text{reduced}}\] If possible
Back-conversion
\[\frac{a}{b} = \lfloor\frac{a}{b}\rfloor\frac{a \bmod b}{b}\] Back to mixed number

Step-by-step Example

Example: Convert 2 6/8 to improper fraction
Step 1: Multiply whole by denominator
\[2 \times 8 = 16\]

Multiply the whole part (2) by the denominator (8).

Step 2: Add numerator
\[16 + 6 = 22\]

Add the result (16) to the original numerator (6).

Step 3: Form improper fraction
\[2\frac{6}{8} = \frac{22}{8}\]

Place the new numerator (22) over the original denominator (8).

Step 4: Simplify
\[\frac{22}{8} = \frac{22 \div 2}{8 \div 2} = \color{blue}{\frac{11}{4}}\]

Reduce the fraction by the gcd (2).

Step 5: Verify
\[\frac{11}{4} = 2.75 = 2\frac{3}{4}\] (equivalent to simplified 2 6/8)
More examples
Simple:
\[1\frac{1}{2} = \frac{1 \cdot 2 + 1}{2} = \frac{3}{2}\]
Larger:
\[3\frac{2}{5} = \frac{3 \cdot 5 + 2}{5} = \frac{17}{5}\]
With simplification:
\[1\frac{4}{6} = \frac{10}{6} = \frac{5}{3}\]
Negative:
\[-2\frac{1}{3} = -\frac{7}{3}\]
Memory aid
Remember formula:
"Whole times denominator, plus numerator"

Example:
3 2/5 → (3×5)+2 = 17
→ 17/5
Negative mixed numbers
\[-2\frac{2}{3}\] is interpreted as \[-(2\frac{2}{3})\]
The sign applies to the whole value.
Conversion steps
1. Whole × denominator
2. + numerator
3. Place over denominator
4. Simplify if possible

An improper fraction always has a numerator greater than or equal to the denominator.

Applications of mixed number conversion

Improper fractions are useful in many mathematical contexts:

Mathematical operations
  • Simplify fraction arithmetic
  • Addition and subtraction
  • Multiplication and division
  • Solving equations
Algebra & Analysis
  • Function representation
  • Limit computations
  • Integral calculus
  • Coordinate systems
Technical applications
  • Gear ratios
  • Scales and proportions
  • CAD constructions
  • Programming
Education & Teaching
  • Develop fraction understanding
  • Extend number sense
  • Algebraic foundations
  • Problem solving strategies

Mathematical context

Description

Converting mixed numbers to improper fractions is a fundamental operation in fraction arithmetic. It simplifies many mathematical procedures because improper fractions are easier to add, subtract, multiply and divide. This transformation shows the equivalence of different representations of rational numbers and is essential for algebraic manipulations and advanced mathematics.

Summary

Improper fractions are preferred for computations while mixed numbers are more intuitive for everyday use. Converting between them is a key tool that provides flexibility when working with rational numbers. This skill underpins advanced mathematical concepts and practical applications in science, engineering and everyday tasks.




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