A complex fraction is a fraction that contains other fractions — for example, one-half over three-quarters. They show up in rates (“3/4 mile in 1/6 hour”), in algebra and in formulas for things like gear ratios. This calculator takes two complex fractions and adds, subtracts, multiplies or divides them, simplifying each one to an ordinary fraction first and showing every step.
How to use the calculator
- For the first complex fraction, enter its numerator fraction and its denominator fraction. Mixed numbers are allowed.
- Choose the operation between the two complex fractions.
- Enter the numerator and denominator of the second complex fraction.
- The tape shows the result as a mixed number, the value of each complex fraction on its own, the simplified fraction and the decimal. The full expression is drawn under the steps.
The method
Step 1: simplify each complex fraction
Divide the top fraction by the bottom fraction by multiplying by the reciprocal of the bottom.
Step 2: combine the results
Now you have two ordinary fractions. Add or subtract them over a common denominator, multiply straight across, or divide by multiplying by the reciprocal. Simplify at the end.
Worked example
Calculate (1/2)/(3/4) + (2/3)/(5/6).
- First complex fraction: 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3.
- Second complex fraction: 2/3 ÷ 5/6 = 2/3 × 6/5 = 12/15 = 4/5.
- Add: 2/3 + 4/5 = 10/15 + 12/15 = 22/15.
- As a mixed number: 1 7/15 (≈ 1.4667).
If the operation were division instead: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
Reading complex fractions in word problems
Complex fractions appear whenever one fractional amount is divided by another:
| Situation | Complex fraction | Value |
|---|---|---|
| Walk 3/4 mile in 1/6 hour — speed? | (3/4)/(1/6) | 4 1/2 mph |
| 2 1/2 cups of flour per 3/4 batch — per batch? | (2 1/2)/(3/4) | 3 1/3 cups |
| 1/3 of a tank used in 1/2 hour — per hour? | (1/3)/(1/2) | 2/3 tank |
Comparing two such rates is exactly an operation on two complex fractions — for instance, subtracting one speed from another.
Tips
- Convert mixed numbers first. 2 1/2 over 3/4 is 5/2 ÷ 3/4.
- Keep the main fraction bar in view. In handwriting, make the main bar longer than the small ones so you know which division comes last.
- Simplify early. Cancelling common factors before multiplying keeps numbers small: 2/3 × 6/5 = (2 × 2)/(1 × 5) = 4/5 after cancelling the 3.
If each part of your complex fraction is itself a sum or difference, such as (1/2 + 1/3)/(3/4 − 1/6), use the simplifying complex fractions calculator. For a single pair of ordinary fractions, the fraction calculator is quicker.
Frequently asked questions
What is a complex fraction?
A complex fraction has a fraction in its numerator, its denominator or both, such as (1/2)/(3/4). It is just a division problem written vertically: (1/2)/(3/4) means 1/2 ÷ 3/4.
How do you add two complex fractions?
Simplify each complex fraction to an ordinary fraction by dividing its top by its bottom, then add the results over a common denominator. (1/2)/(3/4) = 2/3 and (2/3)/(5/6) = 4/5, so the sum is 2/3 + 4/5 = 22/15 = 1 7/15.
Why is the bottom fraction flipped?
Dividing by a fraction is the same as multiplying by its reciprocal. To evaluate (1/2)/(3/4), multiply 1/2 by 4/3 to get 4/6 = 2/3.
What if a denominator is zero?
A complex fraction whose bottom part equals 0 is undefined, just like any division by zero. The calculator tells you which complex fraction is undefined.