Equivalent Fractions
Two fractions may look different but actually represent exactly the
same quantity (or value).
1 3
For example, 1 = .
2 2
Mathematically, the correct way to say that two fractions are equal in
size (or value) is to say that they are equivalent fractions.
Equivalent fractions are fractions that are equal in size.
Note: Two or more fractions that look different (they may have
different numerators and denominators) may still be equivalent.
Example
To demonstrate how two fractions that look different may represent
1
2
and .
the same quantity, consider the fractions
2
4
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As can be seen both of the two fractions above have the same
shaded proportion. Therefore they represent the same quantity.
Hence we can say that
1
2
and are equivalent fractions.
2
4
This is only one example of an equivalent fraction. You will shortly
see that every fraction has an endless list of other fractions that are
equivalent to it.
Creating Equivalent Fractions
Considering the previous example, how do you know exactly which
other fractions are equivalent to a given starting fraction?
Example
Create an equivalent fraction to
3
.
4
To create an equivalent fraction from a given starting fraction you
must follow these steps:
1.
Multiply both the numerator and the denominator of the
fraction by any whole number that you choose:
Let’s choose the number 2.
Multiplying the numerator (3) and the denominator (4) of the
starting fraction by 2 gives:
Numerator:
Denominator:
2.
3!2 = 6
4!2 = 8
The new values (the two answers that you obtained from
multiplying the numerator and denominator by 2) will
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become the new numerator and denominator of the
equivalent fraction:
The new numerator is 6 and the new denominator is 8:
So an equivalent fraction to
3
6
is .
4
8
Remember that this means that these two fractions represent exactly
the quantity (or value) even though they have been written with a
different numerator and denominator.
A little earlier it was stated that every fraction has an endless list of
equivalent fractions. The reason for this is that in step 1 of the
process you could have easily chosen any other whole number such
as 3 (instead of the choice of 2). Remember, that this whole number
was chosen at random - it could have been 3 or 4 or any other whole
number.
If you had chosen another number such as 3 then all this would have
done is generated another equivalent fraction with a different
numerator and denominator. In this case you would have arrived at:
Numerator:
3! 3=9
Denominator:
4 ! 3 = 12
So another equivalent fraction to
3
9
is
.
4
12
In fact you could keep choosing different whole numbers in step 1
such as 4, 5 and so on and generate an endless list of fractions that
are all equivalent to the original fraction:
3 6 9 12 15
= =
=
=
= .....
4 8 12 16 20
Remember that even though all of these fractions look different
(because they all have different numerators and denominators) they
all represent the same quantity. They are equivalent fractions.
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Creating Equivalent Fractions
with Specific Denominators
You have now learnt how to create many equivalent fractions from a
given starting fraction all with different numerators and denominators.
Generally it will only be necessary to create one equivalent fraction
with a specific denominator.
For example, change
2
?
so that it looks like this :
3
12
You don’t know the numerator (the top number) simply because you
have to find its value according to the steps that you just learned in
the previous section but you do know that the denominator must be
12 because this is a requirement of the question.
Recall from the previous section (Creating Equivalent Fractions) that
you can choose any whole number that you want to multiply by the
numerator and denominator to create an equivalent fraction.
In this case, however, you must choose only one specific whole
number instead of choosing a number at random. That number must
be chosen very carefully to satisfy a particular condition. That number
is always chosen so that when you multiply the denominator of the
starting fraction by it you get the denominator of the final fraction that
you require.
Example
Consider the example of making
1.
2 ?
= . To find the numerator follow these steps :
3 12
Multiply both the numerator and the denominator of the
first fraction by a specific number that changes the starting
denominator (in this case 3) to the final denominator (the
denominator of the second fraction which in this case is
12):
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The first fraction (starting fraction) is
2
.
3
In this example you have a starting denominator of 3 which
when multiplied by a certain number must become 12. The only
number that will achieve this is 4. This means that you must
multiply both the numerator (2) and the denominator (3) by 4.
Numerator:
2! 4=8
Denominator:
3 ! 4 = 12
2.
The new values for the numerator and denominator that
you calculated in the previous step become the new
numerator and denominator of the equivalent fraction:
The new numerator is 8 and the new denominator is 12:
This gives an equivalent fraction of
The fraction of
8
.
12
2
8
is equivalent to .
3
12
So as you can see the steps for this procedure are almost identical to
the steps for forming any equivalent fraction that you learned in the
previous section. The only exception is that you are now finding a
specific whole number in step 1 to multiply by the denominator and
numerator rather than a whole number chosen at random.
This procedure is of critical importance in order to carry out the
addition and subtraction of fractions, as you shall shortly see.
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Example
Solve the following:
1 ?
=
4 12
1.
By comparing the denominator of the first fraction (4) with the
denominator of the final fraction (12) you know that 4 must
become 12. For this to happen you must multiply it by 3. The
rule says that you must do exactly the same to the top number
(the numerator) as you have done to the bottom number (the
denominator) and so you must multiply 1 by 3 as well:
Numerator:
1! 3=3
Denominator:
4 ! 3 = 12
2.
So the new numerator is 3 and the denominator is 12:
This gives an equivalent fraction of
The fraction of
3
.
12
1
3
is equivalent to
.
4
12
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