In a mixture, the ratio of the first and second liquid is $2:3$ and in another mixture, the ratio of those two liquids is $5:4$. Find the ratio in which these two mixtures must be mixed so that the quantity of the two liquids in the new mixture becomes equal.

Let us find the fraction of the first liquid in both mixtures:
* In the first mixture, the proportion of the first liquid is $\frac{2}{2+3}=\frac{2}{5}$.
* In the second mixture, the proportion of the first liquid is $\frac{5}{5+4}=\frac{5}{9}$.
* In the final desired mixture, the two liquids must be equal, so the proportion of the first liquid is $\frac{1}{1+1}=\frac{1}{2}$.

Using the rule of alligation:

$$\text{Value 1}=\frac{2}{5},\text{Value 2}=\frac{5}{9},\text{Mean Value}=\frac{1}{2}$$

Calculate the absolute differences:

* Difference 1: $\left|\frac{5}{9}-\frac{1}{2}\right|=\frac{10-9}{18}=\frac{1}{18}$
* Difference 2: $\left|\frac{1}{2}-\frac{2}{5}\right|=\frac{5-4}{10}=\frac{1}{10}$

The required ratio is:

$$\text{Ratio}=\frac{1}{18}:\frac{1}{10}$$

$$\text{Ratio}=10:18=5:9$$