Arithmetic
Improper Fractions vs Mixed Numbers
An improper fraction has a numerator at least as large as its positive denominator. A mixed number writes the same value as a whole number plus a proper fraction.
Reviewed
Two notations for the same quantity
The improper fraction 11/4 means eleven quarter-sized parts. Grouping four quarters into each whole produces two wholes with three quarters left, so 11/4 and 2 3/4 are equivalent.
Improper fractions are usually easier inside formulas because one numerator and one denominator can be multiplied or divided directly. Mixed numbers are often easier to read in measurements and everyday quantities.
a/b = q + r/b, where a = qb + r and 0 <= r < bImproper fraction to mixed number
Divide the numerator by the denominator. The quotient becomes the whole-number part and the remainder becomes the new numerator. Keep the original denominator and reduce the fractional part if necessary.
- Compute the integer quotient q and remainder r.
- Write q r/b.
- Reduce r/b and keep the sign on the complete value.
Mixed number to improper fraction
Multiply the whole number by the denominator, add the fractional numerator, and place the result over the original denominator. A written mixed number such as 3 1/2 means addition, not multiplication.
w n/d = (wd + n)/dWhere this idea is used
- Use improper fractions for algebraic manipulation and fraction multiplication.
- Use mixed numbers when communicating lengths, recipes, and elapsed time.
- Convert before comparing values if the forms make the comparison difficult to see.
Worked examples
Convert 29/6 to a mixed number
Count complete groups of six.
29 = 4 x 6 + 5The quotient is 4 and the remainder is 5
Result: 29/6 = 4 5/6
Convert 3 7/8 to an improper fraction
Combine the three wholes and seven eighths.
3 x 8 = 2424 + 7 = 31
Result: 3 7/8 = 31/8
Common mistakes
- Writing the remainder as the denominator instead of the numerator.
- Adding the whole number directly to the numerator without first multiplying by the denominator.
- Reading 2 1/3 as 2 x 1/3; standard mixed-number notation means 2 + 1/3.
Questions about improper fractions and mixed numbers
Is every improper fraction greater than 1?
With a positive denominator, it is at least 1 when the numerator is at least the denominator. Negative signs require comparing the signed value.
Can a mixed number have an improper fractional part?
It can be written that way, but standard form keeps the fractional part proper and moves complete wholes into the whole-number part.
Which form is more exact?
Both forms are exact and represent the same rational number.