ScientificCalcul

Arithmetic

How to Add and Subtract Fractions

Fractions can be added or subtracted only after their parts have the same size. A common denominator renames each fraction in equal-sized parts before the numerators are combined.

Reviewed

Worked examples

Add 3/4 and 5/6

The least common denominator of 4 and 6 is 12.

  1. 3/4 = 9/12
  2. 5/6 = 10/12
  3. 9/12 + 10/12 = 19/12

Result: 3/4 + 5/6 = 19/12 = 1 7/12

Subtract 7/10 - 1/4

Use 20 as the least common denominator.

  1. 7/10 = 14/20
  2. 1/4 = 5/20
  3. 14/20 - 5/20 = 9/20

Result: 7/10 - 1/4 = 9/20

Why denominators are not added

The denominator tells you the size of each part. One third and one fifth are different-sized parts, just as one meter and one foot are different units. Adding the numerators before converting the units gives no meaningful fraction.

Once both fractions use a common denominator, the part sizes match and only the counts in the numerators need to be combined.

Formulaa/b + c/d = (ad + bc) / bd

Choosing the least common denominator

The product bd is always a common denominator, but the least common multiple of b and d usually keeps the arithmetic smaller. Divide the least common multiple by each denominator to find the factor used to scale that fraction.

  1. Reduce the input fractions if that makes the numbers smaller.
  2. Find L = lcm(b, d).
  3. Rewrite a/b as a(L/b)/L and c/d as c(L/d)/L.
  4. Add or subtract the numerators, then reduce the final fraction.

Mixed numbers and negative fractions

Convert mixed numbers to improper fractions before applying the common-denominator method. Keep subtraction signs with the numerator so the operation is unambiguous.

For subtraction, check the sign of the final numerator before converting back to a mixed number.

Common mistakes

  • Adding denominators, as in 1/3 + 1/5 = 2/8.
  • Scaling a denominator without multiplying its numerator by the same factor.
  • Forgetting to reduce the final result or convert an improper result when a mixed number is requested.

Where this idea is used

  • Combining measurements given in fractional units.
  • Adding probabilities from mutually exclusive outcomes.
  • Working with rational coefficients in algebra.

Questions about adding and subtracting fractions

Do fractions need a common denominator for multiplication?

No. A common denominator is needed for addition and subtraction, not for multiplication or division.

Can I use the product of the denominators?

Yes. It always works for nonzero denominators, although the least common denominator can reduce the amount of arithmetic.

Why should the answer be simplified?

Lowest terms make equivalent answers easier to compare and expose the essential ratio.

Sources and further reading