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Simplifying Complex Fractions Calculator
Instantly simplify any complex fraction to its lowest terms with full step-by-step working. Enter the top and bottom fractions — get the simplified answer and every step explained.
✓Last verified: April 2026 · Sources: Khan Academy, NCTM
Top number of your fraction
Bottom number (leave blank if whole number)
Please enter at least the top numerator value.
Enter 3/4 or just 3 (for whole number)
Top number of your fraction
Bottom number (leave blank if whole number)
Please enter at least the top denominator value.
Enter 5/6 or just 5 (for whole number)
Simplified Result
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Simplified
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Lowest terms
Numerator
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Top
Denominator
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Bottom
Decimal
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Equivalent
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📋 Step-by-Step Working
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Sources & Methodology
✓Fraction simplification uses the standard GCD (Euclidean algorithm) and the divide-by-reciprocal method, as taught in the US Common Core Mathematics Standards (6.NS.A.1) and verified against Khan Academy and NCTM curriculum references.
Khan Academy's fraction division and simplification methodology is the foundational curriculum reference for this calculator's step-by-step working approach.
NCTM Principles and Standards for School Mathematics define the expected methods for simplifying fractions including GCD factoring and the divide-multiply-reciprocal method used here.
US Common Core Standard 6.NS.A.1 defines dividing fractions by fractions using visual models and equations, which underpins the algorithm this calculator implements.
Method: Convert any whole numbers to fractions (n/1). The complex fraction (a/b)/(c/d) is simplified as (a/b) x (d/c) = ad/bc. The result is then reduced to lowest terms by dividing numerator and denominator by GCD(ad, bc), computed using the Euclidean algorithm. If the denominator is negative, signs are adjusted so the denominator is always positive in the final answer.
⏱ Last reviewed: April 2026
How to Simplify Complex Fractions: Two Methods with Examples
A complex fraction is any fraction where the numerator, denominator, or both contain fractions themselves. For example, (3/4) / (5/6) is a complex fraction because both the top and bottom parts are fractions. Simplifying a complex fraction means reducing it to a simple fraction in its lowest terms. There are two standard methods taught in school mathematics: the divide method and the LCD method. Both give the same answer — the divide method is usually faster.
Method 1 — Divide (Flip and Multiply)
(a/b) / (c/d) = (a/b) x (d/c) = ad / bc
Step 1: Keep the numerator fraction as-is. Step 2: Flip the denominator fraction (take its reciprocal). Step 3: Multiply numerator x flipped denominator. Step 4: Simplify the resulting fraction by dividing by GCD.
Example: Simplify (1/2 + 1/3) / (5/6)
LCD of 2, 3, and 6 = 6. Multiply every term by 6:
Numerator: (1/2)x6 + (1/3)x6 = 3 + 2 = 5
Denominator: (5/6)x6 = 5
Result: 5/5 = 1
Common Complex Fraction Examples — Quick Reference
Complex Fraction
After Reciprocal
Multiply
Simplified
Decimal
(1/2) / (3/4)
(1/2) x (4/3)
4/6
2/3
0.6667
(3/4) / (5/6)
(3/4) x (6/5)
18/20
9/10
0.9000
(7/8) / (14/16)
(7/8) x (16/14)
112/112
1
1.0000
(2/3) / (4/9)
(2/3) x (9/4)
18/12
3/2
1.5000
5 / (3/7)
(5/1) x (7/3)
35/3
35/3
11.6667
(2/5) / 4
(2/5) x (1/4)
2/20
1/10
0.1000
How to Simplify a Complex Fraction with Mixed Numbers
When mixed numbers appear in a complex fraction, the first step is always to convert them to improper fractions. A mixed number like 2 and 1/2 becomes 5/2 (multiply the whole number by the denominator and add the numerator: 2x2+1=5, keep denominator 2). Once converted, apply the standard divide method. For example: (2 and 1/2) / (1 and 3/4) = (5/2) / (7/4) = (5/2) x (4/7) = 20/14 = 10/7 = 1 and 3/7.
The Euclidean Algorithm for Finding GCD
To fully simplify a fraction to lowest terms you must find the Greatest Common Divisor (GCD) of the numerator and denominator. The Euclidean algorithm is the most efficient method: divide the larger by the smaller, take the remainder, and repeat. The last non-zero remainder is the GCD. Example: GCD(18,20). 20/18 = 1 remainder 2. 18/2 = 9 remainder 0. GCD = 2. So 18/20 = (18/2)/(20/2) = 9/10. A fraction is in lowest terms when GCD = 1.
💡 Shortcut: Before multiplying across, look for common factors you can cancel first. In (3/4) x (6/5), notice 3 and 6 share factor 3, and 4 and 6 share factor 2. Cancel first: (1/2) x (2/5) = 2/10 ... wait, better: 3 cancels with 6 giving 1 and 2, 4 and 6 share 2 so 4 becomes 2 and 6 becomes 3. Result = (1x3)/(2x5) = 3/10. Cross-cancelling before multiplying keeps numbers smaller and avoids large intermediate steps.
Frequently Asked Questions
Rewrite the complex fraction as the numerator fraction divided by the denominator fraction, then multiply the numerator fraction by the reciprocal of the denominator fraction. Example: (3/4)/(5/6) = (3/4)x(6/5) = 18/20. Divide both by GCD(18,20)=2 to get 9/10. This calculator shows every step automatically.
A complex fraction (also called a compound fraction) is a fraction where the numerator, denominator, or both contain fractions. Examples: (1/2)/(3/4), (2+1/3)/(5/6), or 4/(2/3). Unlike a simple fraction with whole numbers top and bottom, a complex fraction has fractions within fractions. Simplifying means reducing to a single simple fraction in lowest terms.
Keep the top fraction (1/2), flip the bottom fraction to get (4/3), then multiply: (1/2)x(4/3) = 4/6. Find GCD(4,6)=2. Divide both by 2: 4/2=2, 6/2=3. Answer: 2/3. Verify: 2/3 x 3/4 = 6/12 = 1/2. Correct — the result times the denominator fraction should equal the numerator fraction.
Find the LCD of all denominators appearing anywhere in the complex fraction. Multiply every single term (in both numerator and denominator) by this LCD. The fractions cancel out, leaving whole numbers. Simplify the resulting simple fraction. This method is especially useful when the numerator or denominator of the complex fraction is a sum of fractions, like (1/2+1/3)/(5/6) where LCD=6 gives numerator=5, denominator=5, result=1.
Convert all mixed numbers to improper fractions first. Multiply whole part by denominator, add numerator, keep denominator. Example: 2 and 1/2 = (2x2+1)/2 = 5/2. Then apply the divide method: (5/2)/(7/4) = (5/2)x(4/7) = 20/14. GCD(20,14)=2. Answer: 10/7 = 1 and 3/7. Always convert mixed numbers before simplifying.
Dividing by a fraction is equivalent to multiplying by its reciprocal because multiplication and division are inverse operations. (a/b) / (c/d) = (a/b) x (d/c). This works because multiplying (c/d) by its reciprocal (d/c) gives 1, effectively cancelling the division. This rule applies universally to all fractions regardless of what the numerator and denominator are.
Use the Euclidean algorithm: divide the larger by the smaller, take the remainder, replace the larger number with the smaller and the smaller with the remainder. Repeat until the remainder is zero. The last non-zero remainder is the GCD. Example: GCD(18,24). 24/18 = 1 remainder 6. 18/6 = 3 remainder 0. GCD = 6. So 18/24 = 3/4.
A fraction is in lowest terms (fully simplified or fully reduced) when the numerator and denominator share no common factors other than 1, meaning GCD(numerator, denominator) = 1. For example, 6/8 is NOT in lowest terms because GCD(6,8)=2. Dividing both by 2 gives 3/4, which IS in lowest terms because GCD(3,4)=1.
Yes. When the simplified result has a denominator of 1, the fraction equals a whole number. Example: (7/8)/(14/16) = (7/8)x(16/14) = 112/112 = 1. Or (2/3)/(2/9) = (2/3)x(9/2) = 18/6 = 3. Any time the numerator and denominator of the final fraction are equal, the result is 1. Any time the denominator divides evenly into the numerator, the result is a whole number.
Treat the whole number as a fraction with denominator 1. For example, 4 / (2/3): treat 4 as 4/1. Then (4/1) / (2/3) = (4/1)x(3/2) = 12/2 = 6. Or (3/4) / 2: treat 2 as 2/1. Then (3/4) / (2/1) = (3/4)x(1/2) = 3/8. The same flip-and-multiply method works for all cases.
Cross-cancellation means dividing a numerator and a denominator (from different fractions being multiplied) by a common factor before multiplying. This keeps numbers smaller and often avoids simplifying at the end. Example: (3/4) x (8/9). Instead of 24/36 = 2/3, notice 3 and 9 share factor 3, and 4 and 8 share factor 4. Cancel: (1/1)x(2/3) = 2/3. Same answer, smaller numbers throughout.