What are fractions?
Fractions are numbers that
are used to represent parts
of things. The meaning of
a fraction goes like this.
You divide up a whole into
a number of pieces and that
number is the denominator
and it is put at the bottom.
Then you take a number of
those pieces and that number
is called the numerator,
and it is put at the top.
For example in the fraction
3/5, you divide up the whole
into 5 pieces and each piece
is a fifth of the whole.
Then the number 3/5 means
the part of the whole that
you get when you take 3
of those fifths, so it is
called 3 fifths and written
3/5.
You can also think of fractions
as division problems because
you are dividing the number
in the numerator into the
denominator number of equal
pieces and that is exactly
what division does.
Determining Which Fraction
is Biggest
It is good when working
with any kind of number
to have some kind of feel
for how big it is. One way
to get a feel for this with
fractions is to compare
them with each other. When
one fraction is a lot bigger
than the other you can often
do this simply by picturing
how big a piece they refer
to, or you can do some estimating
by asking yourself such
things as whether they are
bigger or smaller than certain
simple fractions like 1/2
or 1/4. If the denominator
is more than twice the numerator
then the fraction is smaller
than 1/2. But if the numbers
are quite close together
this might not be so easy,
so another way to do it
is to put them under a common
denominator, something I
will talk about when I tell
you about adding and subtracting
fraction.
Reducing to Lowest Terms
and Equivalent Fractions
It is possible to write
down many different fractions
that represent the same
sized piece, and such fractions
are called equivalent. You
can do this simply by cutting
up all the pieces into an
equal number of pieces.
The effect of this on the
fraction is to multiply
the numerator and denominator
by the same number, so multiplying
numerator and denominator
by the same number produces
equivalent fractions. For
example if you take 2/3
and cut each of the thirds
into 5 equal pieces, then
the new pieces are 15th,
and it takes 10 of those
pieces to make up the 2/3,
so 2/3 is equivalent to
10/15.
It might help you better
understand this if you draw
some pictures of your own
with different fractions
or maybe for some more fun
try this on a cake some
time. Anyway, once you have
convinced yourself that
this works you see that
it means that you can take
any fraction and write as
many equivalent fractions
to it as you want by multiplying
numerator and denominator
by the same number.
This will be useful later
when we add fractions,
but going the other direction
is more often what you
want to do when you are
writing your final answers
to problems, because it
is best to write fractions
with as small a numbers
as possible. Writing an
equivalent fraction with
the numerator and denominator
as small as possible is
what we mean by reducing
a fraction to lowest terms,
and this is a very important
skill to have. To do this
you must divide out as
large a common factor
as you can until there
are no more common factors
left. Ideally you want
to divide by the greatest
common factor, but if
you don't all you have
to do is reduce some more
until there is nothing
that will go into both
numerator and denominator
evenly anymore.
Addition and Subtraction
When you add fractions with
like denominators you are
adding up various numbers
of pieces of like size.
Remember, the denominator
represents the size of the
pieces and the numerator
represents the number of
pieces. When you are adding
you want to know what the
size that you get when you
put the two sizes together,
so you just add the numerators
to find the total number
of pieces. The size of the
pieces hasn't changed, so
you keep the same denominator.
Similarly with subtraction
you are taking away pieces,
so you just subtract the
numerators, and again the
size of the pieces doesn't
change, so the denominator
is still the same.
The more difficult task
comes when the denominators
are different, because
then you are adding or
subtracting different
sized pieces, so it is
not at all clear what
kind of fraction you can
use to represent the total.
The way to deal with this
is to cut up your pieces
into smaller pieces so
that all the pieces are
the same size. To do this
you need to write the
fractions as equivalent
fractions with a common
denominator. Remember
the only way to write
an equivalent fraction
for a fraction which is
in lowest terms is to
multiply top and bottom
by the same number. This
means that the common
denominator must be a
multiple of all of the
denominators in the problem.
The name for such a creature
is a common multiple.
Any common multiple will
do and one way you can
always find a common multiple
is to multiply the numbers,
but it is best to use
the smallest possible
number which is the least
common multiple, often
abbreviated LCM. See GCFs and
LCMs for help with
finding the LCM.
So to add or subtract
numbers with different
denominators you need
to find the least common
multiple of the denominators
and then write each fraction
as an equivalent fraction
that has that number as
its denominator. To do
that look at each denominator
and ask yourself what
number you need to multiply
the denominator by to
get the LCM and then multiply
the numerator and denominator
by that number. So for
example if your denominators
are 8 and 12, then your
LCM is 24, so you need
to multiply the one with
the denominator of 8 top
and bottom by 3 and the
one with the 12 top and
bottom by 2.
Mixed Numbers
A mixed number represents
a number of wholes plus
a part.
To convert a mixed number
to an improper fraction
you have to convert the
whole number to a fraction
with the same denominator
as the fraction part.
To do that you have to
realize that each whole
is the denominator number
of pieces, so for example
each whole in 4 and 2/3
is 3 thirds. This means
that to figure out how
many fraction pieces there
are in the whole number
part of a mixed number
you need to multiply the
whole number by the denominator.
But then there is also
the fraction part. Putting
this all together, the
denominator is the same
as the denominator of
the fraction part, and
to get the numerator you
multiply the whole number
by the denominator and
then add the numerator.
To convert an improper
fraction to a mixed number
you regard the fraction
as a division and express
the remainder as a fraction.
To express a remainder
as a fraction, the remainder
is the numerator and the
divisor is the denominator.
Addition and Subtraction
of Mixed Numbers
There are two ways to add
and subtract mixed numbers.
One way is to just convert
them into improper fractions
and add or subtract like
usual. This sounds like
the simplest way, and
it is fine to do if the
whole numbers are not
too big like 1 or 2, but
when you have big whole
numbers like 23 or 149,
this will involve a lot
of unnecessary multiplying
of big numbers and there
is a better way.
The better way to do
it, especially for when
you have large whole number
parts is to add or subtract
the fractions and whole
numbers separately and
treat the fraction parts
and the whole number parts
sort of like you treat
the digits of numbers.
For addition this means
that if you add up the
fraction part and it turns
out to be an improper
fraction, then you convert
the fraction part into
a mixed number and carry
the whole number part
of this number to the
ones place of the whole
numbers. For subtraction
it means that sometimes
you will have to borrow.
If the fraction part of
the second number is larger
than that of the first
number then since you
can't subtract a larger
number from a smaller
number you have to borrow
from the ones place of
the whole number part.
So, for example, you might
have to convert 4 and
5/6 into 3 and 11/6. To
do this you would take
one of the wholes and
break it up into 6ths.
1 is 6/6, so you add the
6/6 to the 5/6 to get
11/6. A good quick way
to figure out what the
new fraction part is going
to be when you borrow
this way is to simply
add the numerator and
the denominator. Doing
that in this example would
give you 5+6=11, so the
numerator is 11 just like
I said it was.
Multiplication
Before we learn how to multiply
fractions we ought to
think a bit about what it
means. Addition of fractions
is just combining the pieces
together, but what about
multiplying? Multiplying
whole numbers is repeated
addition, but how do you
repeat something a fraction
number of times? That doesn't
really make sense,
but perhaps another interpretation
of multiplication will help.
If you multiply a fraction
like say 2/3 by a whole
number like 4, one way you
could see it is that you
are making 4 copies of the
2/3, sort of like you were
representing 4 wholes, but
2/3 is the whole.
And when you do that, you
can see from the picture
that you get a total of
8 of those 1/3 pieces, or
the fraction 8/3. Now let's
suppose instead that you
wanted to multiply 2/3 by
4/5. Then you could do it
in the same way representing
4/5, but again thinking
of 2/3 as the whole. I will
make the original whole
bigger so that it will be
easier to see the pieces.
Now we can use this picture
to see how to perform the
operation of fraction multiplication.
Notice that in splitting
up the whole twice, the
total number of pieces that
the whole gets divided into
is the product of the denominators,
3 times 5, or 15, and the
number of pieces we are
taking is the number of
rows times the number of
columns, which is the product
of the numerators. So to
multiply fractions, you
multiply the numerators
and multiply the denominators,
straight across. And no
matter how many fractions
you have to multiply, you
do it the same. Just multiply
all the numerators and all
the denominators, and you'll
do just fine.
Cross Canceling
Cross Canceling is really,
really important to do.
With fractions multiplying
is nicer than adding or
subtracting in many ways,
and one of them is that
you can take care of all
the reducing to lowest terms
before you do the problem.
The reason for this is that
reducing to lowest terms
is taking out common factors
from the numerator and the
denominator, and you can
find this a lot easier before
the multiplying is done.
Because before the multiplying
is done the numerator and
denominator are partly factored,
and you are looking for
common factors. This means
that many times if you multiplied
out first and then tried
to reduce to lowest terms,
you would be multiplying
and then refactoring (unmultiplying)
in order to find the common
factors, and that would
be quite a waste. To avoid
doing this you need to remember
that all of the numerators
will be factors of the numerator
of the product and all of
the denominators will be
factors of the denominator
of the product, so you can
cancel out any factor of
anything in a numerator
with any factor of anything
in a denominator. This is
often called cross canceling,
because if the original
fractions were in lowest
terms, and you only had
two fractions, then all
your canceling would be
on the diagonals, but really
you can cancel anything
upstairs with anything downstairs.
Then if you really have
done all the canceling you
can, after you multiply
your numbers out, the final
answer will automatically
be in lowest terms and you
won't have to do any further
reducing. The only thing
you might have to do with
it further is convert it
to a mixed number if it
is an improper fraction.
Mixed Numbers
Multiplying fractions is
mostly a lot easier than
adding fractions, because
you don't have to mess with
common denominators, so
I suppose it is only justice
that something should be
harder about it. What's
harder about it is that
for multiplying you do absolutely
have to convert your mixed
numbers to improper fractions.
There is no nice way to
multiply mixed numbers.
Multiplying and Adding
It is important to remember
that multiplying fractions
is very different from adding
fractions and not get them
confused. Remember for multiplying
you don't need a common
denominator and you multiply
both the numerators and
the denominators. For adding
you need a common denominator
and you add only the numerators.
Division
Reciprocals
the reciprocal of a number
is the number you can multiply
by it and get 1. So the
reciprocal of 2 is 1/2 because
when you multiply them everything
cross cancels and you get
1. Similarly the reciprocal
of 2/3 is 3/2. When you
multiply any fraction by
the fraction you get by
turning it upside down everything
will cross cancel and you
will get 1. So the reciprocal
of any fraction is the fraction
you get by turning it upside
down. The reciprocal is
also sometimes called the
inverse or more properly
the multiplicative inverse.
How to Divide Fractions
When you divide two fractions
like
you are looking for something
you can multiply 6/7 by
to get 3/4, just like when
you divide
you are looking for something
you can multiply by 8 and
get 56. For
you can get this by multiplying
because look, if you multiply
that by 6/7 you get
and the 7/6 and 6/7 cross
cancel away, and you indeed
get 3/4. So you can turn
all division problems into
multiplication problems
by inverting the second
fraction. So to divide,
multiply by the reciprocal.
Rational Numbers
The rational numbers are
the positive and negative
fractions and whole numbers.
A nice short and precise
way of expressing this is
that a rational number is
any number that can be written
as p/q where p and q are
integers and q is not 0.
This including the integers
themselves, because q can
be 1. Negative fractions
work pretty much the same
as positive fractions. You
just have to follow the
rules of positive and negative
numbers as well as those
for fraction arithmetic.
