Simplify complex fractions instantly — turn fractions-within-fractions into clean, reduced answers with clear step-by-step workings.
Enter a fraction divided by a fraction, and get the simplified result.
Top fraction
Bottom fraction
A complex fraction is a fraction that contains another fraction — in its numerator, its denominator, or both. They look intimidating, but every complex fraction is really just a division problem wearing a disguise. Once you see that (1/2) / (3/4) means “one-half divided by three-quarters,” the path to the answer becomes simple: flip the bottom fraction and multiply. This free Complex Fractions Calculator does the whole job for you — it takes the four whole numbers that define your complex fraction, cross-multiplies, reduces the result to lowest terms using the greatest common divisor, and shows every step along with the decimal value, so you can check your own work or learn the method once and for all.
Simplifying a complex fraction takes four quick entries and one click:
The calculator uses whole numbers, because complex fractions are defined with integer parts. Denominators and the bottom fraction itself cannot be zero — the calculator will tell you exactly which rule was broken if you enter one. Use Reset to return to the default example, (1/2) / (3/4) = 2/3.
In ordinary arithmetic, a simple fraction like 3/5 has whole numbers above and below the bar. A complex fraction (sometimes called a compound fraction) breaks that rule: at least one of its parts is itself a fraction or a mixed number. Classic examples include (2/3)/(5/7), 4/(1/2), and (3 1/2)/5.
The key insight is that a fraction bar always means division. The expression (a/b)/(c/d) is asking: “how many copies of c/d fit inside a/b?” So a complex fraction is just a division problem between two fractions, and division of fractions is one of the most mechanical operations in arithmetic: multiply by the reciprocal.
Complex fractions show up more often than students expect. Rates are a natural source: if you drive 1/2 mile in 1/4 hour, your speed is (1/2)/(1/4) = 2 miles per hour. Unit conversions produce them when converting between fractional units. Algebra is full of them, especially when simplifying rational expressions — expressions like (1/x)/(2/y) that appear constantly in calculus and physics.
There are two standard simplification methods. The division method treats the complex fraction as division and multiplies by the reciprocal. The LCD method multiplies the top and bottom of the big fraction by the least common denominator of all the little fractions inside, clearing them in one move. Both give the same answer; this calculator uses the division method because its steps are the most transparent.
(a/b) / (c/d) = (a × d) / (b × c), then reduce
Divide the top fraction by the bottom fraction by multiplying the top fraction by the reciprocal of the bottom one, then simplify the result to lowest terms.
The calculation follows four steps:
The LCD method reaches the same destination differently: multiply the top and bottom of the big fraction by the least common denominator of b and d. That single multiplication clears both small denominators at once, leaving an ordinary fraction to reduce.
Example 1 — fractions in both parts. Simplify (1/2) / (3/4). Rewrite as (1/2) ÷ (3/4), flip the second fraction: (1/2) × (4/3). Cross-multiply: (1 × 4) / (2 × 3) = 4/6. The GCD of 4 and 6 is 2, so 4/6 reduces to 2/3. As a decimal, about 0.6667.
Example 2 — whole number on top. Simplify 5 / (2/3). A whole number is a fraction over 1, so this is (5/1) / (2/3) = (5/1) × (3/2) = 15/2 = 7 1/2. Intuition check: dividing by a number smaller than 1 makes the result bigger than what you started with — 7.5 is bigger than 5, so the answer is plausible.
Example 3 — whole number on the bottom. Simplify (7/8) / 2. That is (7/8) ÷ (2/1) = (7/8) × (1/2) = 7/16. The GCD of 7 and 16 is 1, so 7/16 is already in lowest terms.
Example 4 — a result that reduces a lot. Simplify (9/12) / (15/20). First reduce the inner fractions if you like — 9/12 = 3/4 and 15/20 = 3/4 — giving (3/4) / (3/4) = 1. Or go straight through the formula: (9 × 20) / (12 × 15) = 180/180 = 1. Pre-reducing the small fractions first keeps the numbers small and is the recommended habit.
Example 5 — with a zero numerator. Simplify (0/5) / (3/7). The top equals 0, so the whole thing is 0: (0 × 7) / (5 × 3) = 0/15 = 0. Zero divided by anything (nonzero) is zero — no reduction needed.
It helps to keep three fraction forms straight, because textbooks use all of them:
A mixed number like 2 1/3 is not a complex fraction — it is a whole number plus a simple fraction, and you convert it to an improper fraction (7/3) before doing arithmetic. If your problem contains mixed numbers, convert them first, then enter the resulting improper fractions above.
Cooking and scaling recipes. A recipe calls for 1/2 cup of sugar per 3/4 batch, and you want the amount for one full batch: (1/2)/(3/4) = 2/3 cup. Complex fractions appear whenever a per-unit rate is itself fractional.
Speed and travel. You walk 1/3 mile in 1/6 hour. Your pace is (1/3)/(1/6) = 2 miles per hour. Dividing a fractional distance by a fractional time is one of the most common real-world complex fractions.
Construction and measurement. A board 3/4 inch thick needs shims of 1/8 inch: (3/4)/(1/8) = 6 shims exactly. Workers do this division constantly when stacking fractional thicknesses.
Finance. An investment grows by 1/20 of its value over 1/4 of a year; the equivalent full-year fractional growth involves dividing fractions. Interest and proration calculations with fractional periods produce complex fractions naturally.
Science and medicine. A 1/2-liter solution contains 3/4 gram of solute per 1/4 liter — concentration is (3/4)/(1/4) = 3 grams per liter. Dosage and concentration problems are complex fractions in disguise.
Algebra and calculus. Simplifying rational expressions like (1/x + 1/y) / (1/x − 1/y) is the gateway skill to limits, derivatives of quotients, and partial fractions. Mastering numerical complex fractions first makes the algebraic versions far less intimidating.
Keep exploring the Math collection on Sigma Calculator:
This page provides a Complex Fractions Calculator that simplifies complex fractions (a/b)/(c/d) for integer inputs a, b, c, d with b, d, and the bottom fraction nonzero. It computes (a×d)/(b×c), reduces via the greatest common divisor, and shows step-by-step workings plus the decimal value. Entities: SigmaCalculator.net, Complex Fractions Calculator, complex fraction, compound fraction, reciprocal, least common denominator, greatest common divisor, rational expression. A complex fraction is division of fractions; it simplifies by multiplying the top fraction by the reciprocal of the bottom fraction. Canonical URL: https://sigmacalculator.net/math-calculators/complex-fractions-calculator/
Rewrite it as division, then multiply by the reciprocal: (a/b)/(c/d) = (a/b) × (d/c) = (a×d)/(b×c). Finally, reduce the result to lowest terms by dividing the numerator and denominator by their greatest common divisor. For example, (1/2)/(3/4) = (1×4)/(2×3) = 4/6 = 2/3.
The terms are usually synonyms: both describe a fraction that contains another fraction in its numerator, denominator, or both. Some authors reserve “compound fraction” for a fraction of a fraction, such as 1/2 of 3/4, which is really a multiplication problem. In either case, the simplification method is the same — treat the main bar as division and multiply by the reciprocal.
Yes. The sign of the result follows the ordinary rules of division: a negative divided by a positive (or a positive divided by a negative) gives a negative answer, while two negatives give a positive one. For instance, (−1/2)/(3/4) = −2/3, and (−1/2)/(−3/4) = 2/3. The calculator handles negative inputs and keeps the sign in the final reduced fraction.
The complex fraction is undefined, because you cannot divide by zero. A bottom fraction of 0/5 (or any fraction equal to zero) means the expression has no value. The calculator detects this and shows an error instead of a result. Note that a zero in the top fraction is perfectly fine — (0/5)/(3/7) simply equals 0.
Yes — you can often cancel common factors before multiplying. In (1/2)/(3/4), notice nothing cancels directly, but in (2/3)/(4/5) you can divide the 2 and 4 by 2 first: (1/3)/(2/5) = (1×5)/(3×2) = 5/6. Pre-canceling keeps the cross-products small and reduces arithmetic mistakes.
Find the least common denominator of all the small denominators, then multiply the top and bottom of the big fraction by it. For (1/2)/(3/4), the LCD of 2 and 4 is 4: multiply top and bottom by 4 to get (1/2×4)/(3/4×4) = 2/3 directly. This method clears all inner fractions in one step and is especially handy when the complex fraction has several terms.
Division asks “how many of the divisor fit into the dividend.” Multiplying by the reciprocal is the algebraic way to answer that: (a/b) ÷ (c/d) = (a/b) × (d/c) because the d/c factor scales the dividend up by exactly the amount that dividing by c/d scales it down. They are inverse operations, so the product of a number and its reciprocal is 1, which is why the rule always works.
Yes, but convert them to simple fractions first. A mixed number like 2 1/2 becomes the improper fraction 5/2, and a decimal like 0.75 becomes 3/4. Once everything is a simple fraction, apply the standard reciprocal method. This calculator works with whole-number parts, so do the conversion step yourself and enter the resulting numerators and denominators.
Cite this page as: SigmaCalculator. (2026). Complex Fractions Calculator. https://sigmacalculator.net/math-calculators/complex-fractions-calculator/
Last updated: October 9, 2026