Once you can read a fraction, a fair question is: what happens when you take more parts than fit in a single whole? If a cake is cut into halves and you have three halves, that is more than one cake — but we can still write it as a single fraction.

A fraction whose top is at least as big as its bottom is called an improper fraction:

The numerator 3 is bigger than the denominator 2, so this is worth more than one whole. Nothing is wrong with it — "improper" just means "top-heavy".

The same amount can be written as a whole number sitting next to a small, proper fraction. That is a mixed number:

Read it as "one and a half": one whole, plus one half left over. A mixed number is really just a hidden addition, 1\tfrac{1}{2} = 1 + \tfrac{1}{2}, but we leave the + out and write the pieces side by side.

The big idea of this page: an improper fraction and a mixed number can name the exact same amount.

See it built

Each bar below is one whole, split into the same number of equal parts. Watch enough parts get shaded to spill past a single whole — then see those parts regroup into one full whole plus a little fraction left over. Step through it.

Converting between them

To turn a mixed number into an improper fraction, multiply the whole by the denominator and add the numerator — that counts how many small parts you have in total. The denominator stays the same:

For example 2\tfrac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}: two whole thirds-bars are 6 thirds, plus the extra 1 third makes 7 thirds.

To go the other way — an improper fraction into a mixed number — divide the top by the bottom. The quotient is the whole number, and the remainder is the new numerator over the same denominator:

See it explained

Sal Khan converts both directions between mixed numbers and improper fractions.