ALGEBRAFractionsMathematics Calculator
🔀

Complex Fractions

A complex fraction has a fraction in its numerator or denominator. Simplify (a/b)/(c/d) by multiplying by (d/c): result = (a×d)/(b×c).

Concept Fundamentals
(a/b)/(c/d) = (a×d)/(b×c)
Rule
(a/b) ÷ (c/d)
As Division
Multiply by d/c
Reciprocal
Reduce to lowest terms
Simplify

Did our AI summary help? Let us know.

Complex fraction = division of two fractions. Multiply top and bottom by LCD of inner fractions to simplify. Result is always a single simplified fraction.

Key quantities
(a/b)/(c/d) = (a×d)/(b×c)
Rule
Key relation
(a/b) ÷ (c/d)
As Division
Key relation
Multiply by d/c
Reciprocal
Key relation
Reduce to lowest terms
Simplify
Key relation

Ready to run the numbers?

Why: Complex fractions appear in algebra, rates, and probability. Simplifying them reveals the underlying ratio.

How: Treat (a/b)/(c/d) as division: multiply numerator by reciprocal of denominator. (a/b) × (d/c) = (a×d)/(b×c).

Complex fraction = division of two fractions.Multiply top and bottom by LCD of inner fractions to simplify.
Sources:Khan Academy — Complex FractionsPurplemath — Complex Fractions

Run the calculator when you are ready.

Complex Fractions CalculatorFraction within fraction

Examples — Click to Load

Enter Complex Fraction (a/b) ÷ (c/d)

Numerator (top) a/b

Denominator (bottom) c/d

Visual: (1/2) ÷ (3/4)

complex-fractions.sh
RESULT
$ simplify (1/2) ÷ (3/4)
Result
2/3
Simplified
2/3
Decimal
0.666667
Mixed
0 2/3
Complex Fractions Calculator
(1/2) ÷ (3/4) = 2/3
Decimal: 0.666667 · Mixed: 0 2/3
numbervibe.com
Share:

Numerator vs Denominator vs Result

Step-by-Step

RULE
Rule(a/b) ÷ (c/d) = (a×d)/(b×c)
MULTIPLY
Substitute(1×4)/(2×3) = 4/6
SIMPLIFY
GCDGCD(4, 6) = 2
RESULT
Result2/3
Decimal0.666667
Mixed0 2/3

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

(a/b)/(c/d) = (a×d)/(b×c)—same as dividing by fraction: multiply by reciprocal.

🔢

Multiply numerator and denominator by b×d to clear inner fractions.

1. Key Takeaways

  • • Complex fraction: (a/b)/(c/d) has a fraction in numerator and/or denominator.
  • • Rule: (a/b) ÷ (c/d) = (a×d)/(b×c) — multiply by reciprocal of divisor.
  • • Always simplify the result to lowest terms using GCD.
  • • Convert to mixed number when numerator ≥ denominator.

2. Did You Know?

Flip and Multiply

Dividing by c/d is the same as multiplying by d/c.

Same Result

(1/2)/(1/2) = 1 — any fraction divided by itself equals 1.

Whole Number Result

(2/3)/(1/6) = 4 — result can be a whole number.

Algebra Connection

Complex fractions appear in rational expressions and algebra.

LCD Alternative

You can also multiply top and bottom by LCD of denominators.

Reciprocal

The reciprocal of a/b is b/a. Their product is 1.

3. How It Works

A complex fraction has a fraction in its numerator, denominator, or both. To simplify (a/b)/(c/d), we use the rule: dividing by a fraction equals multiplying by its reciprocal. So (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). We then simplify the result by dividing numerator and denominator by their GCD.

Inputs

Four integers: numerator fraction (a/b) and denominator fraction (c/d)

Outputs

Result, simplified form, decimal, mixed number

4. Expert Tips

Flip only the divisor

When dividing, flip the denominator fraction (c/d → d/c), not the numerator.

Simplify last

Compute (a×d)/(b×c) first, then simplify. Avoid simplifying intermediate steps unless needed.

Check for zero

The denominator fraction (c/d) must not be zero. If c=0, the fraction is undefined.

Mixed numbers

Convert mixed numbers to improper fractions before applying the rule.

5. Comparison Table

Complex FractionSimplifiedDecimal
(1/2)/(3/4)2/30.667
(2/3)/(1/6)44
(1/3)/(2/3)1/20.5
(1/2)/(1/2)11
(7/8)/(1/4)7/23.5

6. FAQ

What is a complex fraction?

A fraction where the numerator, denominator, or both contain fractions. Example: (1/2)/(3/4).

What is the rule for dividing fractions?

(a/b) ÷ (c/d) = (a×d)/(b×c). Multiply the first by the reciprocal of the second.

Why does (1/2)/(1/2) = 1?

Any number divided by itself equals 1. (1/2)/(1/2) = (1/2)×(2/1) = 2/2 = 1.

Can the result be a whole number?

Yes. (2/3)/(1/6) = (2×6)/(3×1) = 12/3 = 4.

Do I flip both fractions?

No. Only flip the divisor (denominator fraction). The numerator stays as is.

What if the denominator fraction is zero?

Division by zero is undefined. The denominator fraction (c/d) must not equal zero.

How do I handle mixed numbers?

Convert to improper fractions first. E.g. 1½ = 3/2, then apply the rule.

Is (a×d)/(b×c) always in lowest terms?

Not necessarily. Simplify by dividing both by GCD(a×d, b×c) to get lowest terms.

7. Quick Stats

a×d

New numerator

b×c

New denominator

GCD

Simplify

Reciprocal

d/c

8. Sources

Wikipedia — FractionMath Is Fun — Dividing FractionsKhan Academy — Fraction ArithmeticPurplemath — Complex FractionsNCTM — Math StandardsCalculator Soup — Fractions

9. Disclaimer

⚠️ Warning: This calculator is for educational purposes. Complex fraction simplification uses standard mathematical rules. For rigorous applications, verify with authoritative sources.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Decimal to Fraction Converter

Convert decimal numbers to fractions. Handles terminating and repeating decimals with step-by-step solutions.

Mathematics

Equivalent Fractions Calculator

Equivalent Fractions Calculator - Calculate and learn about fractions concepts

Mathematics

Fraction Addition Calculator

Fraction Addition Calculator - Calculate and learn about fractions concepts

Mathematics

Fractions Calculator

Fractions Calculator - Calculate and learn about fractions concepts

Mathematics

Fraction Comparison Calculator

Fraction Comparison Calculator - Calculate and learn about fractions concepts

Mathematics

Fraction Division Calculator

Fraction Division Calculator - Calculate and learn about fractions concepts

Mathematics