Once both parts are reduced, the final step is the one in dividing fractions. Here the calculator works out each part first and then does that division in a single step, with the final quotient and both parts in view.
Worked example: (1/2 + 1/3)/(3/4)
Calculate the complex fraction that has a sum on top and a fraction on the bottom.
- Work out the top. The common denominator is 6, so 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- Work out the bottom. It is already a single fraction: 3/4.
- Divide by the reciprocal. (5/6) ÷ (3/4) = 5/6 × 4/3 = 20/18.
- Simplify. Divide 20 and 18 by 2 to get 10/9, which is 1 1/9 as a mixed number.
Result: 10/9.
Check: 10/9 × 3/4 = 30/36 = 5/6, which is what came out on top.
A complex fraction is a division: the main bar asks how many times the bottom fits into the top. The name refers to the nested structure and has nothing to do with complex numbers. Wikipedia calls it a complex or compound fraction.
LCD shortcut for working by hand
There is another route that avoids the reciprocal. Multiply the top and the bottom by the LCM of all the inner denominators and the fractions disappear. With (1/5 + 3/10)/(1/4 − 1/20), the LCM of 5, 10, 4 and 20 is 20. The top becomes 4 + 6 = 10 and the bottom 5 − 1 = 4, so the quotient is 10/4 = 5/2.
The calculator does not use the shortcut: it evaluates each part and divides, with the same final value. Dividing by a fraction is the same as multiplying by its reciprocal; the guide to the reciprocal of a fraction explains why.
Forms it can take
The main bar can have fractions, whole numbers or expressions on either side. Dividing by something less than 1 makes the absolute value of the result larger, and dividing by something greater than 1 makes it smaller.
| Form | Top | Bottom | Result |
|---|---|---|---|
| Fraction over whole number: (5/8)/2 | 5/8 | 2 | 5/16 |
| Whole number over fraction: 3/(2/5) | 3 | 2/5 | 15/2 |
| One over a sum: 1/(1/4 + 1/6) | 1 | 5/12 | 12/5 |
| Negative result: (1/10 − 1/2)/(1/5) | −2/5 | 1/5 | −2 |
The sign of the quotient comes from the signs of the two parts. In the last row the top is negative and the bottom is positive, so the result is −2. If both parts were negative, the quotient would be positive again.
When it gives an error
It gives an error when the bottom equals zero. Typing 1/(2/3 − 2/3) produces a zero denominator after calculating, and then the final division does not exist. A division by zero inside either field is rejected too. A part such as 0/5 is valid on top, but it cannot be the divisor.
Letters are not accepted: a fraction with x needs domain rules, and that is what the algebraic fractions calculator is for. To split a quotient of polynomials into separate terms, use partial fractions. For a multi-level fraction with a name of its own, see continued fractions.
Where they show up
Two real situations use it. Two resistors of 4 Ω and 6 Ω in parallel are equivalent to 1/(1/4 + 1/6) = 12/5 Ω, as in the table, and the article on parallel resistors explains why a sum of reciprocals appears.
The other is the average speed of a round trip. If you drive out at 60 mph and come back at 40 mph over the same road, the average is 2/(1/60 + 1/40) = 2/(1/24) = 48 mph, not 50. The article on average speed for a round trip develops it.
When the expression is a long line with no clear main bar, the fraction expression calculator follows the order of operations, and to divide a fraction by a whole number you can also multiply fractions by the reciprocal.
Source: Wikipedia, article Fraction, section “Complex and compound fractions”, which defines a fraction whose numerator or denominator is a fraction or a mixed number (accessed October 17, 2026).
Frequently asked questions
What happens if the mixed number is negative?
The sign applies to the whole number, not just the whole-number part. (−2 1/3)/(7/9) reads as (−7/3)/(7/9) = −7/3 × 9/7 = −3. Put the sign in front of the mixed number, with a space between the whole part and the fraction, and the result keeps the sign.
How is a chain of bars like 3/4/6 read?
From left to right: (3/4)/6 = 1/8. If you want the 6 to divide the 4 first, write 3/(4/6) = 9/2. The same thing happens when you copy the line from a spreadsheet or a word processor.