There are two drums, each containing a mixture of paints $A$ and $B$ . In drum $1, \mathrm{~A}$ and B are in the ratio $18: 7$. The mixtures from drums $1$ and $2$ are mixed in the ratio $3: 4$ and in this final mixture, $A$ and $B$ are in the ratio $13: 7$. In drum $2$, then $A$ and $B$ were in the ratio

We need to find the ratio of paints A and B in drum 2 using alligation.

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Drum 1: A and B are in ratio 18:7

This means out of every 25 parts, 18 parts are A and 7 parts are B

So fraction of A in drum 1 = $\frac{18}{25}$

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Final mixture: A and B are in ratio 13:7

So fraction of A in final mixture = $\frac{13}{20}$

Mixing ratio: Drum 1 and Drum 2 are mixed in ratio 3:4

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Let us call the fraction of A in drum 2 as k (this is what we need to find).

Alligation is a method to find the ratio of mixing when we know the concentrations. The key formula is:

$\frac{\text{Concentration of A} - \text{Final concentration}}{\text{Final concentration} - \text{Concentration of B}} = \frac{\text{Quantity of B}}{\text{Quantity of A}}$

In our case:

$\frac{\frac{18}{25} - \frac{13}{20}}{\frac{13}{20} - k} = \frac{4}{3}$

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Calculate $\frac{18}{25} - \frac{13}{20}$

To subtract these fractions, we need a common denominator:

$\frac{18}{25} = \frac{18 \times 4}{25 \times 4} = \frac{72}{100}$

$\frac{13}{20} = \frac{13 \times 5}{20 \times 5} = \frac{65}{100}$

So: $\frac{18}{25} - \frac{13}{20} = \frac{72}{100} - \frac{65}{100} = \frac{7}{100}$

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Substitute back into our equation:

$\frac{\frac{7}{100}}{\frac{13}{20} - k} = \frac{4}{3}$

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Cross multiply:

$3 \times \frac{7}{100} = 4 \times \left(\frac{13}{20} - k\right)$

$\frac{21}{100} = 4 \times \frac{13}{20} - 4k$

$\frac{21}{100} = \frac{52}{20} - 4k$

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Convert to common denominator:

$\frac{21}{100} = \frac{260}{100} - 4k$

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Solve for k:

$4k = \frac{260}{100} - \frac{21}{100} = \frac{239}{100}$

$k = \frac{239}{400}$

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Now we know that the fraction of A in drum 2 is $\frac{239}{400}$.

The fraction of B in drum 2 is: $1 - \frac{239}{400} = \frac{400-239}{400} = \frac{161}{400}$

Therefore, the ratio of A to B in drum 2 is:

$\frac{239}{400} : \frac{161}{400} = 239 : 161$

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A and B are in the ratio 239:161 in drum 2.