Simplifying Complex Fractions Calculator: Divide & Reduce

How simplifying a complex fraction works

A complex fraction is a fraction with a fraction in it — one fraction divided by another. The one here is (1/2) ÷ (3/4): the top fraction is 1/2 and the bottom fraction is 3/4. The rule for dividing fractions is to flip the second fraction and multiply: (1/2) ÷ (3/4) becomes (1/2) × (4/3), because the reciprocal of 3/4 is 4/3. Multiply straight across — top times top, bottom times bottom — to get (1 × 4)/(2 × 3) = 4/6. Then reduce: 4 and 6 share a greatest common divisor of 2, so dividing both by 2 gives 2/3. That is the whole method in three moves: flip, multiply, reduce. This calculator divides one fraction by another; our fraction simplifier reduces a single fraction — a different job, and a different tool.

Two things about complex fractions surprise people. The first is that the answer can be bigger than both fractions you started with: (3/4) ÷ (1/2) = 3/2, because dividing by a fraction smaller than 1 is the same as multiplying by a number bigger than 1. The second is that dividing a fraction by itself always gives 1: (5/6) ÷ (5/6) = 30/30 = 1. Negatives work the same way — the sign stays on the numerator, so (-1/2) ÷ (3/4) = -4/6 = -2/3. And the reverse direction is just as useful: if you know the result and the bottom fraction, the top fraction is the result times the bottom fraction. A result of 2/3 over a bottom fraction of 3/4 means the top fraction was (2/3) × (3/4) = 6/12 = 1/2.

The worked example, line by line

Setup: top fraction 1/2; bottom fraction 3/4.

Line 1 — flip the bottom fraction. The reciprocal of 3/4 is 4/3.

Line 2 — multiply across. (1/2) × (4/3) = (1 × 4)/(2 × 3) = 4/6.

Line 3 — find the GCD. The greatest common divisor of 4 and 6 is 2.

Line 4 — reduce. 4 / 2 = 2 and 6 / 2 = 3, so the answer is 2/3.

Line 5 — check as a decimal. 2 / 3 = 0.6667, and 0.5 / 0.75 = 0.6667 — the same number, so the answer checks out.

Line 6 — the reverse. A result of 2/3 over a bottom fraction of 3/4 means the top fraction was (2 × 3)/(3 × 4) = 6/12 = 1/2.

Every branch, checked

(2/3) ÷ (4/5): flip to (2/3) × (5/4) = (2 × 5)/(3 × 4) = 10/12, and the GCD of 10 and 12 is 2, so the answer is 5/6.

(3/4) ÷ (1/2): flip to (3/4) × (2/1) = (3 × 2)/(4 × 1) = 6/4, and the GCD of 6 and 4 is 2, so the answer is 3/2 — bigger than either fraction you started with, because the bottom fraction was smaller than 1.

(5/6) ÷ (5/6): flip to (5/6) × (6/5) = (5 × 6)/(6 × 5) = 30/30, and the GCD of 30 and 30 is 30, so the answer is 1. Any fraction divided by itself is 1.

(7/8) ÷ (2/3): flip to (7/8) × (3/2) = (7 × 3)/(8 × 2) = 21/16, and the GCD of 21 and 16 is 1, so it is already in lowest terms.

(-1/2) ÷ (3/4): flip to (-1/2) × (4/3) = (-1 × 4)/(2 × 3) = -4/6, and the GCD of 4 and 6 is 2, so the answer is -2/3. The sign stays on the numerator.

Reverse — finding the top fraction: a result of 2/3 over a bottom fraction of 3/4 means the top fraction was (2 × 3)/(3 × 4) = 6/12 = 1/2. The top fraction is always the result times the bottom fraction.

What the defaults assume

All four inputs are whole numbers: the calculator works with exact integer arithmetic, so there is no rounding anywhere except the decimal check, which is shown to four places. A denominator of zero is not allowed in either fraction — a fraction with zero on the bottom is undefined — and a bottom fraction of zero is not allowed either, because dividing by zero is undefined. Negative numbers are allowed in the numerators and are carried on the numerator of the answer; a negative denominator is flipped to a positive one with the sign moved to the numerator, which is the same fraction written the usual way. Mixed numbers are not accepted: convert a mixed number like 1 1/2 to the improper fraction 3/2 first. The result is always shown in lowest terms, and a whole-number result is shown as a plain number, not as a fraction over 1.

Output · formula · meaning

Who uses this

• The middle-school student: has (1/2) ÷ (3/4) for homework and learns the answer is 2/3, with every step shown — flip to 4/3, multiply to 4/6, reduce by 2 — so the method sticks for the test, not just the answer.
• The teacher: needs worked examples for a lesson on dividing fractions and finds (3/4) ÷ (1/2) = 3/2 — the perfect counter-example to the idea that division always makes things smaller.
• The exam-prep student: is drilling fraction division and learns that (7/8) ÷ (2/3) = 21/16 with a GCD of 1 — so some answers arrive already reduced, and checking the GCD first saves a step.
• The baker scaling a recipe: needs half of three-quarters of a cup and learns (1/2) ÷ (3/4) is not the operation at all — but dividing a batch by a fraction uses the same flip-and-multiply, and the calculator shows the working either way.

Accuracy & limitations

The math is exact: the calculator uses whole-number arithmetic, so the cross-multiplied fraction and its GCD are computed without any rounding, and the result is always in lowest terms. The decimal is the only rounded number, shown to four places for checking. The calculator accepts whole numbers only — decimals and mixed numbers must be converted to improper fractions first. A zero denominator in either fraction, or a bottom fraction equal to zero, is rejected with an error, because both are undefined in mathematics. Very large inputs (over 100,000 in size) are rejected to keep the arithmetic exact. If you need to simplify a single fraction rather than divide two of them, use our fraction simplifier instead. Read the add fractions calculator or the divide fractions calculator for the neighbouring operations.