Fractions can be added or subtracted only after their parts refer to the same-sized pieces. That is why unlike denominators must be converted to a common denominator.
Method
- Find the least common denominator.
- Rewrite each fraction as an equivalent fraction.
- Add or subtract the numerators.
- Keep the common denominator.
- Simplify using the greatest common factor.
Worked addition
For 3/4 + 5/6, the least common denominator is 12. Convert 3/4 to 9/12 and 5/6 to 10/12. The sum is 19/12, or 1 7/12.
Worked subtraction
For 7/8 − 1/3, use denominator 24. The fractions become 21/24 and 8/24. The result is 13/24, already in simplest form.
Why cross-multiplication works
The general formula a/b + c/d = (ad + bc)/bd creates a common denominator. Using the least common denominator usually keeps numbers smaller, but both methods produce an equivalent result after simplification.
Use the Scientific Calculator to check decimal equivalents and the Percentage Calculator for percentage conversions.
Frequently asked questions
Can I add denominators directly?
No. Adding 1/2 + 1/2 gives 2/2, not 2/4.
What is an improper fraction?
Its numerator is at least as large as its denominator. It can be converted to a mixed number.
When is a fraction simplified?
When numerator and denominator have no common factor greater than one.
Choosing the least common denominator
For denominators 8 and 12, list multiples until the first shared value: 8, 16, 24 and 12, 24. The least common denominator is 24. Using 96 would also work, but it creates larger numerators and requires more simplification.
To check an answer, convert both the original expression and the final fraction to decimals. For 3/4 + 5/6, 0.75 + 0.8333… is about 1.5833…, matching 19/12. The decimal check can reveal arithmetic errors, while the fraction remains the exact result.
Check with estimation
Before calculating exactly, estimate the answer using familiar benchmark fractions. Because 3/4 is 0.75 and 5/6 is a little more than 0.8, their sum should be between 1.5 and 1.6. An exact answer outside that range signals a conversion or arithmetic error.