We will take our chocolate bar to understand this.

Let us say you have two such chocolate bars and two friends to share them with. You divide the first chocolate into $18$ small tablets and give $15$ of them to your first friend. Then you divide the second chocolate into $18$ tablets and give $14$ of them to the second friend. Clearly you have given more to your first friend.

Let us understand this using fractions. To your first friend you gave $\dfrac{\cancel{15} \raise{5pt}{5}}{\cancel{18} \lower{5pt}{6}} = \dfrac{5}{6}$ of one chocolate. To your second friend you have given $\dfrac{\cancel{14} \raise{5pt}{7}}{\cancel{18} \lower{5pt}{9}} = \dfrac{7}{9}$ of one chocolate. Since you already know that you have given more to you first friend, we can say that:

$\dfrac{5}{6} \gt \dfrac{7}{9}$

Let us try another one, with the same two chocolate you divide the first one into $3$ equal parts, and give your first friend $2$ of them, so you have given your first friend $12$ small tablets. You divide the other chocolate into $9$ equal parts, and give your second friend, $2$ of them, so you have given your second friend $4$ small tablets. In terms of fractions, you have given $\dfrac{2}{3}$ of a chocolate to your first friend, while to your second friend you have given $\dfrac{2}{9}$ of a chocolate. But you know, that you have given more quantity to your first friend as compared to your second friend. So we can say:

$\dfrac{2}{3} \gt \dfrac{2}{9}$

If you are comparing fractions with the same numerator, and different denominators, then the fraction with the larger denominator is smaller. It is easy to understand this if you remember the larger number of pieces you divide anything into, smaller each piece will be.

So, when you are comparing two fractions, you can compare them by making either the denominators same or the numerators same. We will take an example of each.

Let us compare the fractions $\dfrac{7}{13}$ and $\dfrac{15}{26}$

Here we see that, if we make the numerators same we have to multiply the first fraction by $15$ in both the numerator and denominator and the second fraction by $7$. So our fractions will become,

$\dfrac{7 \times 15}{13 \times 15}$ and $\dfrac{15 \times 7}{26 \times 7}$. These are quite large multiplications.

Instead, let us see what do we need to do if we were to make the denominators same. All we need to do is multiply the first fraction by $2$ in the numerator and the denominator.

So, our fractions become:

$\dfrac{7 \times 2}{13 \times 2}$ and $\dfrac{15}{26}$

which is:

$\dfrac{14}{26}$  and $\dfrac{15}{26}$

Now, we have both the denominators same and the numerator of the first fraction larger than the second fraction.

So, we know that $\dfrac{14}{26} \lt \dfrac{15}{26}$ and that means:

$\dfrac{7}{13} \lt \dfrac{15}{26}$

Now let us try one more. This time we will compare the fractions $\dfrac{8}{11}$ and $\dfrac{12}{13}$.

If we try to make the denominators same like before we will need to multiply the first fraction by $13$ and the second fraction by $11$ both in the numerator and the denominator. So our fractions will become:

$\dfrac{8 \times 13}{11 \times 13}$ and $\dfrac{12 \times 11}{13 \times 11}$ and that will give us quite large values requiring large calculations. Let us see if there is an easier way to do this. What if we try to make the numerators same?

We can see that the LCM of the numerators is $24$ and the numerators can be made same by multiplying the first fraction by $3$ and the second fraction by $2$.

So, our fractions become:

$\dfrac{8 \times 3}{11 \times 3}$ and $\dfrac{12 \times 2}{13 \times 2}$ , that is

$\dfrac{24}{33}$ and $\dfrac{24}{26}$

We know that when the numerators are same, the fraction with the smaller denominator is the larger fraction.

Therefore:

$\dfrac{24}{33} \lt \dfrac{24}{26}$ that means, using our original fractions we get:

$\dfrac{8}{11} \lt \dfrac{12}{13}$