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 $=$.

Learn the trick:
If the numerators are identical, the larger fraction is the one with the smaller denominator!

Why does this make sense?
Look at the following example:
Dana’s mom made a pizza.
Dana invited Jonathan for dinner, so the pizza was shared equally between Dana and Jonathan.
Each received half a pizza: Dana got $\frac{1}{2}$​, and Jonathan got $\frac{1}{2}$​.

Now, imagine Dana invited Bar and Ofir for dinner as well.
The pizza would need to be shared among four children: Dana, Jonathan, Bar, and Ofir.
The pizza would then be divided into $\frac{1}{4}$​, and each child would get $\frac{1}{4}$​​.

Obviously, when more children share the pizza, each person gets less.
The more we divide a whole into pieces, the smaller each piece becomes.

This means that if the numerators are identical but the denominators are different,
the largest fraction will be the one with the smallest denominator.

Exercises:
Mark the correct sign >, <:

$3 \over5$_____________$3 \over4$

Mark the correct sign >, <:

$6 \over100$_____________$6 \over100$

Sometimes, you can compare fractions by comparing them to $1$, $\frac{1}{2}​$, and $\frac{1}{3}$​.

How do you compare a fraction to \(1\)?

If the numerator is larger than the denominator, the fraction is greater than$1$.
If the numerator is smaller than the denominator, the fraction is smaller than $1$.

For the number $1$, the numerator is equal to the denominator.
Therefore, if the numerator and denominator are equal, the fraction equals $1$.

Exercise:

Mark the correct sign >, <:

$3 \over5$_____________$5 \over4$

Solution:
In the fraction $\frac{3}{5}$, the numerator is smaller than the denominator. Hence, the fraction is less than $1$.
In the fraction $\frac{5}{4}$​, the numerator is larger than the denominator. Hence, the fraction is greater than $1$.
If one fraction is less than $1$ and the other is greater than $1$, the larger fraction is the one greater than $1$.

Exercise:

Mark the correct sign >, <:

$11 \over12$_____________$3 \over1$

Solution:
In the fraction $\frac{11}{12}$​, the numerator is smaller than the denominator. Hence, the fraction is less than $1$.
In the fraction $\frac{3}{1}$​, the numerator is larger than the denominator. Hence, the fraction is greater than $1$.

How do you compare a fraction to $\frac{1}{2}$​?

Look at the fraction $\frac{1}{2}$​.
In this fraction, the denominator is twice the numerator.
Similarly, any fraction equal to $\frac{1}{2}$ will have a denominator twice as large as the numerator.

Therefore, to compare a fraction to $\frac{1}{2}$: Multiply the numerator by $2$.

• If the result is greater than the original denominator, the fraction is greater than $\frac{1}{2}$​.
• If the result is less than the original denominator, the fraction is less than $\frac{1}{2}$​.

Exercise:

Mark the correct sign >, <:

$2 \over5$_____________$6 \over9$

Solution:
Look at the fraction $\frac{2}{5}$.
Multiply the numerator by $2$ to get $4$.
$4$ is less than the original denominator $5$, so the fraction is less than $\frac{1}{2}$.

Now look at the fraction $\frac{6}{9}$.
Multiply the numerator by $2$ to get $12$.
$12$ is greater than the original denominator $9$, so the fraction is greater than $\frac{1}{2}$​.

If one fraction is less than $\frac{1}{2}$​ and the other is greater than $\frac{1}{2}$​, the larger fraction is the one greater than $\frac{1}{2}$​.

How do you compare a fraction to $\frac{1}{3}​$​?

Look at the fraction $\frac{1}{3}​$​.
In this fraction, the denominator is three times the numerator.
Similarly, any fraction equal to $\frac{1}{3}​$ will have a denominator three times as large as the numerator.

Therefore, to compare a fraction to $\frac{1}{3}​$​: Multiply the numerator by $3$.

• If the result is greater than the original denominator, the fraction is greater than $\frac{1}{3}​$​.
• If the result is less than the original denominator, the fraction is less than $\frac{1}{3}​$​.

Exercise:

Mark the correct sign >, <:

$1 \over6$_____________$4 \over5$

Solution:
Look at the fraction $4 \over5$​.
Multiply the numerator by $3$ to get $12$.
$12$ is greater than the original denominator $5$, so the fraction is greater than $\frac{1}{3}​$​​.

Now look at the fraction $\frac{1}{6}​$​​.
Multiply the numerator by $3$ to get $3$.
$3$ is less than the original denominator $6$, so the fraction is less than $\frac{1}{3}​$​​.