What a complex fraction is
A complex fraction is a fraction whose top, bottom or both contain fractions themselves, like (2/3)/(5/6) or (1/2 + 1/3)/(3/4 − 1/6). They turn up in algebra, in rates such as miles per ¾ hour, and in recipes being scaled. This calculator simplifies them to a single fraction in lowest terms and shows the working, so you can follow or check each line.
How to simplify a complex fraction
The fraction bar means "divide", so a complex fraction is really one division. There are two standard methods, and the calculator shows both whenever they apply.
Method 1: simplify, then divide. Simplify the top to a single fraction, simplify the bottom to a single fraction, then divide: keep the top, change ÷ to ×, and flip the bottom.
Method 2: multiply by the LCD. Find the least common denominator of every small fraction, and multiply the top and the bottom by it. All the small fractions clear at once, leaving whole numbers.
Worked example: (1/2 + 1/3) ÷ (3/4 − 1/6)
- Top: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
- Bottom: 3/4 − 1/6 = 9/12 − 2/12 = 7/12
- Divide: 5/6 ÷ 7/12 = 5/6 × 12/7 = 60/42 = 10/7, or 1 3/7
With the LCD method: the denominators are 2, 3, 4 and 6, so the LCD is 12. The top becomes 6 + 4 = 10 and the bottom 9 − 2 = 7, giving 10/7 directly. OpenStax works the similar example (1/2 + 2/3)/(3/4 − 1/6) and gets 7/6 ÷ 7/12 = 2; try it above.
Nested fractions
A fraction can sit inside another fraction's denominator, as in continued fractions. Work from the inside out: 1 + 1/2 = 3/2, then 1/(3/2) = 2/3, then 1 + 2/3 = 5/3, and finally 1 ÷ 5/3 = 3/5. Type the bottom as 1 + 1/(1 + 1/2) to see these steps.
| Complex fraction | Simplified |
|---|---|
| (1/2)/(3/4) | 2/3 |
| (2/3)/(5/6) | 4/5 |
| (2 1/4)/(3/8) | 6 |
| (1/2 + 1/3)/(3/4 − 1/6) | 10/7 |
| 1/(1 + 1/(1 + 1/2)) | 3/5 |
Limits
- Numbers only. For complex rational expressions with x in them, simplify by hand with the same LCD method.
- If the bottom works out to 0, the complex fraction has no value, and the calculator says so.
- Decimals are turned into exact fractions first, so 0.5 is treated as 1/2.