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

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

Why adding across the fraction bar fails

Incorrect: 1/2 + 1/3 = 2/5

Correct: 1/2 + 1/3 = 3/6 + 2/6 = 5/6

The denominator names the size of each piece. Halves and thirds must be renamed as equal-sized sixths before their counts can be added.

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.

Worked example 1

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

Choose a common denominator

Use the least common multiple when it is easy to see; any common multiple works, but may require extra reduction.

1. Compute 5/12 + 7/18.

Hint: LCM(12, 18) = 36.

Answer: 15/36 + 14/36 = 29/36

2. Compute 2/3 - 5/6.

Hint: Rename 2/3 as 4/6.

Answer: 4/6 - 5/6 = -1/6

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.

Denominator and sign checks

  • 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.

Sources and further reading