Complex Fractions Calculator

Simplify complex fractions containing fractions in the numerator or denominator. Results in lowest terms.

Simplify complex fractions where one fraction is divided by another. Get step-by-step solutions with results in lowest terms.

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help_outlineHow to Useexpand_more

Working with Complex Fractions

A complex fraction has a fraction in either the numerator, the denominator, or both. This calculator simplifies complex fractions by converting them to simple fractions in their lowest terms.

The Simplification Process:

  1. Identify the top (numerator) and bottom (denominator) fractions
  2. Find the reciprocal of the bottom fraction (flip it)
  3. Multiply the top fraction by this reciprocal
  4. Reduce the result to lowest terms

Mixed Numbers and Integers:

Complex fractions may also include mixed numbers or simple integers in either position. Convert mixed numbers to improper fractions first, and treat integers as fractions over 1.

infoWhat is Complex Fractions Calculator?expand_more

A complex fraction (also called a compound fraction) is a fraction where the numerator, denominator, or both contain fractions themselves. For instance, (1/2)/(3/4) is a complex fraction with 1/2 as the numerator and 3/4 as the denominator.

Complex fractions appear frequently in algebra, calculus, and real-world applications like rate problems, unit conversions, and scaling calculations. Understanding how to simplify them is essential for working with ratios and proportions.

The key to simplifying any complex fraction is recognizing that dividing by a fraction is the same as multiplying by its reciprocal. This transforms the complex fraction into a standard multiplication problem.

functionsFormulaexpand_more

Complex Fraction Formula:

(a/b) ÷ (c/d) = (a/b) × (d/c)

= (a × d) / (b × c)

The Reciprocal Rule:

To divide by a fraction, multiply by its reciprocal.

The reciprocal of c/d is d/c (the fraction flipped).

With Integers:

n ÷ (a/b) = n × (b/a) = (n × b) / a

(a/b) ÷ n = (a/b) × (1/n) = a / (b × n)

lightbulbExamplesexpand_more

Example 1: (1/2) ÷ (3/4)

Multiply by reciprocal: (1/2) × (4/3)

= (1 × 4) / (2 × 3) = 4/6

Simplified: 4/6 = 2/3

Example 2: (3/5) ÷ (2/7)

Multiply by reciprocal: (3/5) × (7/2)

= (3 × 7) / (5 × 2) = 21/10

Result: 21/10 = 2 1/10

Example 3: (5/8) ÷ (5/4)

Multiply by reciprocal: (5/8) × (4/5)

= (5 × 4) / (8 × 5) = 20/40

Simplified: 20/40 = 1/2

quizFAQexpand_more
What makes a fraction 'complex'?expand_more
A fraction is complex when it contains another fraction in its numerator, denominator, or both. It's essentially a fraction within a fraction, or a division of two fractions stacked vertically.
How do I simplify a complex fraction step by step?expand_more
First, identify the top and bottom fractions. Then multiply the top fraction by the reciprocal (flipped version) of the bottom fraction. Finally, multiply the numerators together and denominators together, then reduce to lowest terms.
What if there's an integer in a complex fraction?expand_more
Treat any integer as a fraction over 1. For example, if you have 3 ÷ (2/5), rewrite 3 as 3/1, then multiply: (3/1) × (5/2) = 15/2. Similarly, (2/5) ÷ 3 becomes (2/5) × (1/3) = 2/15.
Can complex fractions contain mixed numbers?expand_more
Yes! First convert any mixed numbers to improper fractions. For example, 1 1/2 becomes 3/2. Then proceed with the normal simplification process using the reciprocal method.
Why is multiplying by the reciprocal the same as dividing?expand_more
Division and multiplication by reciprocals are inverse operations. When you divide by a number, you're asking 'how many times does this fit?' Multiplying by the reciprocal gives the same answer because you're essentially scaling in the opposite direction.