Mixed Numbers and Improper Fractions
A recipe asks for one and a half cups of flour, a bottle holds two and a quarter litres, a
plank is three and a half metres long — real amounts often come as a whole plus a bit left
over. That "whole plus a bit" is a mixed number.
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:
\frac{3}{2}
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". Two halves make one whole cake, and there is still
one more half left over.
The same amount can be written as a whole number sitting next to a small, proper fraction.
That is a mixed number:
1\tfrac{1}{2}
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.
\frac{3}{2} = 1\tfrac{1}{2}
Suppose every pizza is cut into 4 equal slices, and you eat
9 slices. That is the improper fraction
\tfrac{9}{4} — nine quarter-slices. Four slices rebuild one
whole pizza, and four more rebuild a second whole pizza; that uses up
8 slices and leaves 1 slice over.
So \tfrac{9}{4} = 2\tfrac{1}{4}: two whole pizzas and
one slice. The mixed number is just the tidy way to say how many whole pizzas
and how many loose slices you have.
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.
See it as round pizzas
Here is the same idea drawn as round pizzas. Each pizza is split into the same number of
equal slices, and a random number of slices is shaded — always more than one whole pizza's
worth. Count the completely filled pizzas, then the leftover
slices in the last one: that is the mixed number. Press Refresh
for a fresh fraction.
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:
w\,\tfrac{a}{b} = \frac{w \times b + a}{b}
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:
\frac{7}{3} = 2\tfrac{1}{3} \quad\text{since}\quad 7 \div 3 = 2 \text{ remainder } 1
Three worked examples
Read each one slowly — the trick is always "how many wholes fit, and what is left over?"
-
\tfrac{5}{2} — five halves. Two halves make a whole, and they
do that twice (using 4 halves), leaving 1 half.
So \tfrac{5}{2} = 2\tfrac{1}{2} because
5 \div 2 = 2 \text{ r } 1.
-
\tfrac{9}{4} — nine quarters. Four quarters make a whole,
twice (8 quarters), leaving 1 quarter. So
\tfrac{9}{4} = 2\tfrac{1}{4}.
-
Going back the other way: 3\tfrac{2}{5}. Three wholes are
3 \times 5 = 15 fifths, plus 2 more
makes 17. So
3\tfrac{2}{5} = \tfrac{17}{5}.
Two whole cakes and a third cake cut into thirds with one slice gone — what's left? That
is 2\tfrac{2}{3} cakes: two full cakes plus two thirds of the
last one. As an improper fraction it is \tfrac{8}{3}, because
2 \times 3 + 2 = 8 thirds altogether. Mixed numbers are how we
usually talk about cake ("two and two-thirds"), and improper fractions are how we
usually calculate with it.
On the number line
A mixed number tells you exactly where to stand on the number line. The whole number says
which two whole numbers you are between, and the fraction says how far along you
go. 2\tfrac{1}{2} sits between 2 and
3, exactly halfway. Because
\tfrac{5}{2} = 2\tfrac{1}{2}, the improper fraction lands on the
very same spot — same point, two names.
Two traps that catch a lot of people:
-
\tfrac{7}{2} is three and a half
(3\tfrac{1}{2}, since 7 \div 2 = 3 \text{ r } 1)
— it is not "seven point two". A fraction bar is a division, not a
decimal point.
-
An improper fraction is not "wrong". "Improper" only means top-heavy;
\tfrac{5}{2} is a perfectly correct number. In fact when you
multiply or divide fractions, the improper form is usually the
easier one to use.
See it explained
Sal Khan converts both directions between mixed numbers and improper fractions.