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**Hint:**We are given two rational numbers and we need to find out the greatest rational number between them. For that, we will Take LCM of all the rational numbers and after that compare its numerator. The rational number with a greater numerator will be the greatest rational number.

**Complete step by step solution:**

In this question, we are given two rational numbers and 4 more rational numbers between them and we need to find out which one of these rational numbers is the greatest one.

The given rational numbers are: $\dfrac{3}{7}$ and $\dfrac{9}{{13}}$

And the rational numbers between them are: (1) $\dfrac{4}{{11}}$ (2) $\dfrac{7}{{12}}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{5}{8}$

To check which rational number is greater between the given two numbers, if the denominator of the numbers is not same, then we need to take the LCM and then compare the numerator.

So, the numbers are $\dfrac{4}{{11}}$, $\dfrac{7}{{12}}$ , $\dfrac{1}{2}$and $\dfrac{5}{8}$.

So, we need to find the LCM of 11, 12, 2 and 8.

The LCM of 11, 12, 2 and 8 is 264.

Therefore, multiply and divide $\dfrac{4}{{11}}$ with 24, $\dfrac{7}{{12}}$ with 22 , $\dfrac{1}{2}$ with 132 and $\dfrac{5}{8}$ with 33, we get

$ \Rightarrow \dfrac{4}{{11}} \times \dfrac{{24}}{{24}} = \dfrac{{96}}{{264}}$

$ \Rightarrow \dfrac{7}{{12}} \times \dfrac{{22}}{{22}} = \dfrac{{154}}{{264}}$

$ \Rightarrow \dfrac{1}{2} \times \dfrac{{132}}{{132}} = \dfrac{{132}}{{264}}$

$ \Rightarrow \dfrac{5}{8} \times \dfrac{{33}}{{33}} = \dfrac{{165}}{{264}}$

Now, we have rational numbers with same denominator: $\dfrac{{96}}{{264}}$, $\dfrac{{154}}{{264}}$, $\dfrac{{132}}{{264}}$ and $\dfrac{{165}}{{264}}$.

So, here we can compare the numerator now. Therefore, here 165 is greater than 96, 154 and 132.

Hence, $\dfrac{{165}}{{264}}$ is the greatest rational number between $\dfrac{3}{7}$ and $\dfrac{9}{{13}}$.

**Therefore, we can say that $\dfrac{5}{8}$ is the greatest rational number between $\dfrac{3}{7}$ and $\dfrac{9}{{13}}$.**

**Note:**

Here, we can also check directly which one is greatest by dividing the given numbers. Therefore, we get

$ \Rightarrow \dfrac{4}{{11}} = 0.36$

$ \Rightarrow \dfrac{7}{{12}} = 0.58$

$ \Rightarrow \dfrac{1}{2} = 0.50$

$ \Rightarrow \dfrac{5}{8} = 0.62$

Here, 0.62 is greater than 0.36, 0.58 and 0.50. Hence, we can say that $\dfrac{5}{8}$ is the greatest rational number between $\dfrac{3}{7}$ and $\dfrac{9}{{13}}$.

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