Simplifying Complex Fractions Calculator reduces a fraction stacked over another fraction into a single simplified fraction, step by step. It rewrites the vertical division as a horizontal one, multiplies by the reciprocal, and reduces the product to lowest terms.
Recognize a complex fraction
A complex fraction has a fraction in its numerator, its denominator, or both, such as (3/4) written over (5/8). The horizontal bar separating the two levels represents division, exactly the same as a division sign written between (3/4) and (5/8). Simplifying Complex Fractions Calculator treats every complex fraction as a division problem from the first step onward.
Rewrite as a single division
The rewrite step turns (a/b) over (c/d) into (a/b) ÷ (c/d). This is not a shortcut or an approximation; it is the definition of what a stacked fraction bar means. Simplifying Complex Fractions Calculator displays this rewrite explicitly because skipping it is where students most often lose track of which fraction needs to be flipped in the next step.
Multiply by the reciprocal
Dividing by a fraction is the same as multiplying by its reciprocal, so (a/b) ÷ (c/d) becomes (a/b) × (d/c). Take (3/4) over (5/8): rewrite as (3/4) ÷ (5/8), then multiply by the reciprocal: (3/4) × (8/5). Multiply numerators: 3 × 8 = 24. Multiply denominators: 4 × 5 = 20. The unreduced result is 24/20.
Reduce the result to lowest terms
24/20 shares a common factor of 4 between numerator and denominator: 24 ÷ 4 = 6, and 20 ÷ 4 = 5, giving the simplified fraction 6/5, or 1 and 1/5 as a mixed number. Simplifying Complex Fractions Calculator always performs this final reduction automatically, since an unreduced fraction is not considered a fully simplified answer.
Simplify a complex fraction with addition inside a level
Some complex fractions have a sum or difference inside one level instead of a single fraction, such as (1/2 + 1/3) over (5/6). The first task is to combine the top into a single fraction before treating the whole thing as a complex fraction: 1/2 + 1/3 becomes 3/6 + 2/6, which is 5/6.
The complex fraction becomes (5/6) over (5/6), which simplifies to 1, since dividing any nonzero fraction by itself gives 1.
Avoid this common mistake
A frequent error tries to cancel terms diagonally across the complex fraction before rewriting it as a division, which only works correctly if the whole numerator and the whole denominator are each single, unadded fractions. When either level contains a sum, combine that level into one fraction first; canceling individual terms before combining produces a wrong result almost every time.
Simplify a complex fraction with a whole number level
A complex fraction can also have a plain whole number stacked over a fraction, or a fraction stacked over a plain whole number, and both cases reduce to the same reciprocal method by treating the whole number as a fraction over 1.
Take 5 over (2/3): rewrite 5 as 5/1, then the complex fraction becomes (5/1) ÷ (2/3), which becomes (5/1) × (3/2) = 15/2, or 7 and 1/2 as a mixed number. Simplifying Complex Fractions Calculator treats a whole-number level as an implicit fraction over 1 automatically, so no separate rule is needed for this case.
Check the result against decimal division
Converting both fractions in a complex fraction to decimals before dividing gives an independent way to check the fraction-based answer. For (3/4) over (5/8), 3/4 = 0.75 and 5/8 = 0.625, and 0.75 ÷ 0.625 = 1.2, which matches 6/5 converted to a decimal exactly.
This decimal cross-check is especially useful for confirming complex fractions that involve larger or less familiar numbers, where a reciprocal-multiplication error is easy to make but a decimal mismatch is easy to spot.
Frequently asked questions
How do you simplify a complex fraction?
To simplify a complex fraction, rewrite the fraction bar as a division sign between the top and bottom fractions, multiply the top fraction by the reciprocal of the bottom fraction, then reduce the resulting fraction to lowest terms.
What is (3/4) divided by (5/8)?
(3/4) divided by (5/8) equals 6/5, found by multiplying (3/4) by the reciprocal of (5/8), which is (8/5), giving 24/20 before reducing to 6/5.
What do you do if a complex fraction has addition inside it?
If a complex fraction has addition or subtraction inside one level, combine that level into a single fraction using a common denominator first, before treating the whole expression as a division of two fractions.
Can you cancel terms across a complex fraction before simplifying?
Canceling terms across a complex fraction before simplifying is only valid once each level is a single fraction with no addition or subtraction remaining; canceling individual terms inside a sum produces an incorrect result.
Is a complex fraction the same as a compound fraction?
Complex fraction and compound fraction are often used interchangeably to describe a fraction containing another fraction, though usage varies slightly by textbook. Simplifying Complex Fractions Calculator handles the standard stacked-division structure regardless of which term is used.
How do you check that a complex fraction simplification is correct?
To check a complex fraction simplification, convert the original complex fraction and the simplified answer both to decimals and confirm they match. (3/4) over (5/8) as a decimal is 0.75 / 0.625 = 1.2, matching 6/5 = 1.2 exactly.
How do you simplify a whole number stacked over a fraction?
To simplify a whole number stacked over a fraction, rewrite the whole number as itself over 1, then apply the standard reciprocal method: 5 over (2/3) becomes (5/1) × (3/2) = 15/2.
What is 5 divided by 2/3?
5 divided by 2/3 equals 15/2, or 7 and 1/2 as a mixed number, found by multiplying 5 by the reciprocal of 2/3, which is 3/2.
Summary
Simplifying Complex Fractions Calculator reduces a fraction stacked over another fraction by rewriting the structure as a division, multiplying by the reciprocal, and reducing the result to lowest terms. When a level contains a sum or difference, combine it into a single fraction first before applying the reciprocal step.
Enter any complex fraction to see each of these stages worked out in order.