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. See
How to
Add, Subtract, Multiply, and Divide
Integers for information about the
integers. Negative fractions work pretty
much the same as positive fractions.
You just have to follow the rules of
positive and negative numbers (See
How to
Add, Subtract, Multiply, and Divide
Integers ) as well as those for
fraction arithmetic.
More Examples and Practice
The
MathHelp
collection of problem sets
Rational
Numbers will give you some more
examples and practice.
