In an alloy, zinc and copper are in the ratio 1 : 2. In the second alloy, the same elements are in the ratio 2 : 3. If these two alloys be mixed to form a new alloy in which two elements are in the ratio 5 : 8, the ratio of these two alloys in the new alloys is -

Let them be mixed in the ratio x : y
Then, in 1^st alloy, Zinc = $$\frac{x}{3}$$ and Copper = $$\frac{{2x}}{3}$$
2^nd alloy, Zinc = $$\frac{{2y}}{5}$$ and Copper = $$\frac{{3y}}{5}$$
Now, we have
$$\eqalign{ & \frac{x}{3} + \frac{{2y}}{5}:\frac{{2x}}{3} + \frac{{3y}}{5} = 5:8 \cr & {\text{or,}}\frac{{5x + 6y}}{{10x + 9y}} = \frac{5}{8} \cr & {\text{or,}}40x + 48y = 50x + 45y \cr & {\text{or,}}10x = 3y \cr & \therefore \frac{x}{y} = \frac{3}{{10}} \cr} $$
Thus, the required ratio = 3 : 10

Alligation Method :
You must know that we can apply this rule over the fractional value of either zinc or copper.
Let us consider the fractional value of zinc.

Therefore, they should be mixed in the ratio
$$\eqalign{ & = \frac{1}{{65}}:\frac{2}{{39}} \cr & {\text{or,}}\frac{1}{{65}} \times \frac{{39}}{2} \cr & = \frac{3}{{10}} \cr & {\text{or,}}\,\,3:10 \cr} $$