In an alloy 'A', zinc and copper are present in the ratio $1 : 1$. In another alloy 'B', the ratio of zinc to copper is $3 : 5$. These two alloys are mixed to form a new alloy where the ratio of zinc to copper is $2 : 3$. Find the ratio in which alloy 'A' and alloy 'B' are mixed.

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**Solution:**

1. Let the quantities of alloys A and B mixed be $x$ and $y$ respectively.

2. In alloy A, zinc and copper are in the ratio $1 : 1$, so the fraction of zinc is $\frac{1}{1+1} = \frac{1}{2}$.

3. In alloy B, zinc and copper are in the ratio $3 : 5$, so the fraction of zinc is $\frac{3}{3+5} = \frac{3}{8}$.

4. In the resulting alloy, zinc and copper are in the ratio $2 : 3$, so the fraction of zinc is $\frac{2}{2+3} = \frac{2}{5}$.

5. Using the mixture formula for zinc fraction:
   $$\frac{x 	imes \frac{1}{2} + y 	imes \frac{3}{8}}{x + y} = \frac{2}{5}$$

6. Multiply both sides by $(x + y)$:
   $$x 	imes \frac{1}{2} + y 	imes \frac{3}{8} = \frac{2}{5} (x + y)$$

7. Multiply through by 40 (LCM of denominators 2, 8, and 5) to clear denominators:
   $$40 	imes \left(x 	imes \frac{1}{2} + y 	imes \frac{3}{8}ight) = 40 	imes \frac{2}{5} (x + y)$$

$$20x + 15y = 16(x + y)$$

8. Expand right side:
   $$20x + 15y = 16x + 16y$$

9. Rearrange terms:
   $$20x - 16x = 16y - 15y$$

$$4x = y$$

10. Therefore, the ratio of alloy A to alloy B is:
    $$x : y = 1 : 4$$

**Answer:** $1 : 4$

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