Complex Fraction Calculator simplifies a fraction stacked over another fraction, in the form (a/b)/(c/d), by rewriting the division as a multiplication by the reciprocal. Enter the top fraction and the bottom fraction, and the tool returns a single simplified fraction along with mixed and decimal forms.
Rewrite division as multiplication
A complex fraction is a division problem written vertically instead of with a division sign. (a/b)/(c/d) means the same thing as (a/b) ÷ (c/d), which by the standard rule for dividing fractions equals (a/b) × (d/c).
Complex Fraction Calculator performs this rewrite as its first documented step, since recognizing the stacked fraction as a division is the key insight that unlocks the rest of the simplification.
Work through a full example
Take (2/3) over (4/5). Rewrite as division: (2/3) ÷ (4/5). Flip the second fraction and multiply: (2/3) × (5/4). Multiply numerators: 2 × 5 = 10. Multiply denominators: 3 × 4 = 12. The result is 10/12, which reduces by the common factor 2 to 5/6.
Complex Fraction Calculator shows every one of these lines, from the initial rewrite through the final reduction.
Handle a complex fraction with a whole number
A complex fraction does not require both levels to be fractions; one side can be a whole number written as a fraction over 1. Take 3 over (2/7). Rewrite 3 as 3/1, then divide: (3/1) ÷ (2/7) = (3/1) × (7/2) = 21/2, which is 10 and 1/2 as a mixed number.
Complex Fraction Calculator accepts a bare whole number on either level and converts it internally before applying the reciprocal step.
Handle a negative complex fraction
Signs in a complex fraction follow the same rule as any fraction division: the result is negative when exactly one of the two fractions is negative, and positive when both share the same sign. (−2/3) over (4/5) becomes (−2/3) × (5/4), giving −10/12, which reduces to −5/6.
Complex Fraction Calculator tracks the sign through the reciprocal step rather than requiring it to be resolved separately beforehand.
Simplify a complex fraction with a sum inside a level
Some complex fractions have an addition or subtraction inside one of the two levels rather than a single clean fraction. Take (1/2 + 1/4) over (3/8). First combine the top into a single fraction using a common denominator: 1/2 + 1/4 becomes 2/4 + 1/4, which is 3/4.
Now the complex fraction is (3/4) over (3/8), a standard two-fraction division: (3/4) × (8/3) = 24/12, which reduces to 2. Complex Fraction Calculator handles this combine-first step whenever a level contains more than one term.
Avoid this common mistake
The most common error multiplies straight across without flipping the second fraction, treating (a/b)/(c/d) as (a×c)/(b×d) instead of (a×d)/(b×c). For (2/3) over (4/5), this mistake would give (2×4)/(3×5) = 8/15, which is wrong; the correct reciprocal-based result is 5/6. Always flip the bottom fraction before multiplying, never the top one.
Convert the result to a decimal for a quick check
Converting both the original complex fraction and the simplified answer to decimals offers an independent check on the work. (2/3) over (4/5) as decimals is 0.6667 divided by 0.8, which is 0.8333, matching 5/6 converted to a decimal exactly.
This decimal cross-check is a fast way to catch a reciprocal error, since forgetting to flip the second fraction typically produces a noticeably different decimal value rather than a subtly wrong one.
Recognize a complex fraction inside a larger expression
Complex fractions often appear nested inside a larger algebraic expression rather than standing alone, such as within a rate calculation like miles per hour when both the distance and time are themselves fractions.
A trip covering 3/4 of a mile in 1/6 of an hour has a speed of (3/4) divided by (1/6), which becomes (3/4) × (6/1) = 18/4, reducing to 4.5 miles per hour.
Recognizing this rate-style division as a complex fraction, rather than treating the numbers as unrelated, makes the reciprocal method directly applicable.
Frequently asked questions
What is a complex fraction?
A complex fraction is a fraction that has another fraction in its numerator, its denominator, or both, written in the stacked form (a/b)/(c/d). It represents a division of two fractions.
How do you simplify a complex fraction?
To simplify a complex fraction, rewrite it as a division problem, flip the bottom fraction to its reciprocal, multiply the two fractions straight across, and reduce the result to lowest terms.
What is (2/3) divided by (4/5)?
(2/3) divided by (4/5) equals 5/6, found by multiplying (2/3) by the reciprocal of (4/5), which is (5/4), giving 10/12 before reducing to 5/6.
Can a complex fraction have a whole number in it?
A complex fraction can have a whole number in either level, treated as that whole number written over 1. A whole number of 3 over a fraction of 2/7 becomes (3/1) divided by (2/7).
What is the difference between a complex fraction and a compound fraction?
Complex fraction and compound fraction generally describe the same structure, a fraction containing another fraction, though some textbooks reserve "compound fraction" for a mixed number times a fraction instead. Complex Fraction Calculator handles the stacked-division form described here.
How do you know if a complex fraction is fully simplified?
A complex fraction result is fully simplified once it is written as a single fraction whose numerator and denominator share no common factor greater than 1. Reduce by dividing both by their greatest common divisor to confirm.
What do you do if one level of a complex fraction has addition in it?
If one level of a complex fraction has addition or subtraction in it, combine that level into a single fraction using a common denominator first, before applying the reciprocal-and-multiply rule to the whole expression.
Is (2/3)/(4/5) the same as 2/3 divided by 4/5?
(2/3)/(4/5) written as a complex fraction means exactly the same thing as 2/3 divided by 4/5 written with a division sign; the stacked fraction bar and the division sign are two notations for the identical operation.
How do you check a complex fraction simplification with decimals?
To check a complex fraction simplification with decimals, convert the top and bottom fractions to decimals and divide them directly, then confirm the result matches the decimal value of the simplified fraction answer.
Can a complex fraction appear in a rate problem?
A complex fraction can appear in a rate problem whenever both the numerator quantity and the denominator quantity are themselves fractions, such as computing speed from a fractional distance traveled in a fractional amount of time.
Summary
Complex Fraction Calculator simplifies (a/b)/(c/d) by converting the stacked division into a multiplication by the reciprocal of the bottom fraction, then reducing the product to lowest terms. Enter any two fractions, including whole numbers or negative values, to see the rewrite, the multiplication, and the simplified result together with mixed and decimal equivalents.