By Lisa Maloney | Updated Aug 30, 2022
Improper fractions—where the numerator exceeds the denominator—are essentially hidden mixed numbers. When adding or subtracting them, it’s best to keep them in improper form until the final step, at which point you can convert to a mixed number if desired.
The procedure mirrors that for proper fractions.
Ensure both fractions share the same denominator. If they don’t, adjust one or both by multiplying by a fraction equivalent to 1. For example:
\(\frac{5}{4} + \frac{13}{12}\)
Since 4 × 3 = 12, multiply \(\frac{5}{4}\) by \(\frac{3}{3}\):
\(\frac{5}{4} × \frac{3}{3} = \frac{15}{12}\)
Now the fractions are \(\frac{15}{12}\) and \(\frac{13}{12}\).
With a common denominator, simply add the numerators:
\(15 + 13 = 28\)
Result: \(\frac{28}{12}\)
Reduce the fraction to lowest terms: \(\frac{28}{12} = \frac{7}{3}\). Then, if you wish, express it as a mixed number:
7 ÷ 3 = 2 remainder 1 → \(2 \tfrac{1}{3}\).
Subtracting follows the same steps.
If the denominators differ, first find a common one.
Keep the order of the numbers. For example:
\(\frac{6}{4} – \frac{5}{4}\)
Subtract numerators: 6 – 5 = 1. The result is \(\frac{1}{4}\).
Here \(\frac{1}{4}\) is already in simplest form, and since it is no longer improper, no mixed‑number conversion is required.
When a mixed number is involved, convert it to an improper fraction first:
2 \(\tfrac{1}{6}\) + \(\tfrac{8}{6}\)
Convert the mixed number: 2 × \(\tfrac{6}{6}\) = \(\tfrac{12}{6}\). Add the remaining \(\tfrac{1}{6}\) to get \(\tfrac{13}{6}\).
Now add: \(\tfrac{13}{6} + \tfrac{8}{6} = \tfrac{21}{6}\).
Convert back to a mixed number: \(\tfrac{21}{6} = 3 \tfrac{3}{6}\). Simplify the fractional part to \(\tfrac{1}{2}\), yielding the final answer:
2 \(\tfrac{1}{6}\) + \(\tfrac{8}{6}\) = 3 \(\tfrac{1}{2}\).
