Comparing Fractions
Sometimes we need to compare two fractions to discover which is larger or smaller.
There are two easy ways to compare fractions: using decimals; or using the same
denominator
The Decimal Method of Comparing Fractions
Just convert each fraction to decimals, and then compare the decimals.
Example: which is bigger: 3/8 or 5/12 ?
You need to convert each fraction to a decimal. You can do this using your calculator
(3÷8 and 5÷12).
Anyway, these are the answers I get:
3
/8 = 0.375, and 5/12 = 0.4166...
So, 5/12 is bigger.
The Same Denominator Method
The denominator is
the bottom number
in a fraction.
It shows how many
equal parts the item
is divided into
If two fractions have the same denominator then they are easy to compare:
Example:
4
/9 is less than 5/9 (because 4 is less than 5)
But if the denominators are not the same you need to make them the
same (using Equivalent Fractions).
Example: Which is larger: 3/8 or 5/12 ?
If you multiply 8 × 3 you get 24 , and if you multiply 12 × 2 you also get 24, so let's
try that (important: what you do to the bottom, you must also do to the top):
3
8
× 3
= × 3
9
24
and
It is now easy to see that 9/24 is smaller than
so /12 is the larger fraction.
5
5
12
10
× 2
= × 2
10
24
/24, (because 9 is smaller than 10).
How to Make the Denominators the Same
The trick is to find the Least Common Multiple of the two denominators. In the
previous example, the Least Common Multiple of 8 and 12 was 24.
Then it is just a matter of changing each fraction to make it's denominator the Least
Common Multiple.
Example: Which is larger: 5/6 or
13
/15?
The Least Common Multiple of 6 and 15 is 30. So, let's do some multiplying to make
each denominator equal to 30 :
5
6
× 5
= × 5
25
so
and
30
Now we can easily see that
13
13
15
26
× 2
= × 2
26
30
/30 is the larger fraction
/15 is the larger fraction.
Why a Common Denominator
The denominator can be thought of as a “unit” rather than a number. So ¾ is just 3
“quarters”, like you have 3 “dogs” or 3 “apples” or 3 “children” etc. And just as 4
dogs is more than 3 dogs, 4 “quarters” is more than 3 “quarters”.
But is 3 dogs more than 1 elephant? Well, that would depend on what me mean be
“more”. In fractions though, by “more” we just mean “more of the whole”.
And it is easy to compare which of 2 fractions represents “more of the whole” if the
pieces in each fraction are the same!!!
Other Tricks #1 : If the Numerators are the Same
If you have the same numerator, then it is ALSO easy to compare. For example, what
is more.
½ or 1/3 or ¼?
Well, in each case we have 1 of something. In one case, that something is a “half”, in
the other, it is “a third”, and in the last one, it is a “quarter”. Now, the denominator
tells us how many parts we cut the whole into, so the larger this number is, the
smaller our parts. So the largest fraction out of the three will have the smallest
numerator. So ½ is largest.
HOWEVER: THIS ONLY WORKS IF AND ONLY IF the numerators are the
SAME!!!!!!!!! It is a common mistake to use this method without ensuring the
numerators are the same.
NOTE: Sometimes it is actually better to get a common numerator rather than a
common denominator
Example:
Compare 5/19 to 10/37 and state what is bigger.
Best Solution:
Note, it is easy to make the numerators BOTH 10 than it is to
get a common denominator between 19 and 37.
So change the first fraction to 10/38 (equivalent).
Now we compare 10/38 to 10/37. In both cases we have 10,
but thirty-sevenths are bigger than thirty-eighths, so we say
10/37th must be bigger (greater fraction of a whole).
Other Tricks #2 : Compare everything relative to one half
A very quick way to compare fractions is to quickly think “roughly” where that
fraction would be on the number line, and by “roughly” I really just mean will it be
ABOVE or BELOW ½.
It is very easy to determine this… if the numerator is at least half as big as the
denominator, then clearly the fraction is more than ½.
Examples:
4/7 (4 is more than 3.5 which is half of 7)
12/22 (12 is more than 11 which is half of 22)
So just half the denominator, and see if the numerator is above or below it.
Examples:
8/18. Half of 18 is 9. 8 is less than it. So this fraction is less than half.
13/20. Half of 20 is 10. 13 is above this. So the fraction is more than ½.
So without worrying about HCF, I know 13/20 is larger than 8/18!!!
When comparing fractions, dividing a group up into 2 (those bigger than ½ and those
less than ½) can be useful.
For example: Compare ¾, 4/5, 7/10, 1/3, and 2/5.
Ok, comparing 5 fractions the traditional way will require putting them all under a
common denominator, in this the HCF(4,5,10,3) is 60.
Instead, a quick look tells us that 1/3 and 2/5 are the only 2 less than half. We can
make them both over 15 to get 5/15 and 6/15. So 1/3 is smallest, and 2/5 is 2nd
smallest.
Next we only need to compare ¾, 4/5 and 7/10. 20 is the common denominator. So
we get 15/20, 16/20, and 14/20. So we get 7/10 is 3rd smallest, ¾ is 4th smallest,
and 4/5 is largest. You may not like this way but for some people it is better than
getting a common denominator for a large number of fractions.