Comparing Fractions

To compare fractions, all we need to do is find a common denominator – that is, bring both fractions to a state where the denominators are the same.
Then, we compare the numerators. The fraction with the larger numerator will be the greater one.

• If there is a mixed fraction or a whole number - we will convert them to improper fractions and only then find a common denominator.

A common denominator will be – the product of the denominators! (Remember that we multiply both the numerator and the denominator)
Sometimes, we won't need to multiply the denominators and can perform an operation on just one fraction (expansion or reduction – in cases where one fraction already has the common denominator in the denominator).
In any case, we need to bring both fractions to the same denominator.

Let's practice:
Mark the correct sign $>,<,=$

Solution:
Let's look at the two fractions and notice that $4$ is the common denominator.
Since $4$ is in the denominator of one of the fractions, we will need to multiply the numerator and denominator of the other fraction - $1 \over 2$ by $2$
We get:
$3 \over 4$______________$2 \over 4$

Now, since the denominators are the same, we can just compare the numerators to find out which is greater. Since $3$ is greater than $2$, the sign will be $>$.

Another exercise:
Mark the correct sign $>,<,=$

Solution:
We can immediately see that it is $1=1$ and that the fractions are identical (any number divided by itself equals $1$), but we will still follow the method and find a common denominator.
The common denominator will be $4$ and we get:
$4 \over 4$_____________$4 \over 4$

Now it can be clearly seen that the fractions are identical.

Mark the correct sign $>,<,=$

Solution:
Find a common denominator by multiplying the denominators.
Multiply the fraction $3 \over 4$ by $5$, the denominator of the second fraction, and the fraction $5 \over 4$.
Multiply by $4$, the denominator of the other fraction.
Remember! – Multiply both the numerator and the denominator.
We get:
$15 \over 20$___________________$16 \over 20$

Now we compare based on the numerators only. $16$ is greater than $15$ and therefore the sign is $<$

Another exercise:
Mark the correct sign $>,<,=$

$1 \frac{1}{2}$_____________________$2\frac{2}{6}$

Solution:
Since we see mixed numbers in the exercise, we automatically convert them to improper fractions.

$2\frac{2}{6}=\frac{14}{6}$
$1\frac{1}{2}=\frac{3}{2}$

Let's rewrite the exercise:

We find a common denominator by multiplying the denominators and we get:
$28 \over 12$___________________$18 \over 12$
We compare the numerators, and therefore the sign will be $>$

Note - if you found a different common denominator – for example 6, that's perfectly fine and as long as you found it correctly, it's not a mistake.

Another exercise:
Mark the correct sign $>,<,=$

$3$_____________________$6 \over 2$

Solution:
We will convert the whole number $3$ into a fraction - $3 \over 1$ and rewrite the exercise:

Now we will find the common denominator 2 and we get:

$6 \over 2$_____________________$6 \over 2$
The fractions are identical, so the sign will be $=$.