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Two types of tea, $\text{A}$ and $\text{B}$, are mixed and then sold at Rs. $40$ per kg. The profit is $10\%$ if $\text{A}$ and $\text{B}$ are mixed in the ratio $3 : 2$, and $5\%$ if this ratio is $2 : 3$. The cost prices, per kg, of $\text{A}$ and $\text{B}$ are in the ratio

1. $18:25$
2. $19:24$
3. $21:25$
4. $17:25$

Let the cost price per kg of type $A,$ and $B$ be $\text{Rs.} x$ and $\text{Rs.} y$ respectively.

When, $A$ and $B$ are mixed in the ratio of $3:2,$ then the profit is $10\%,$

So, selling price $:$

$\left( \frac{3x+2y}{5} \right) \times \frac{110}{100} = 40$

$\Rightarrow \left( \frac{3x+2y}{5} \right) \times \frac{11}{10} = 40 \quad \longrightarrow (1)$

When, $A$ and $B$ are mixed in the ratio of $2:3,$ then the profit is $5\%.$

So, selling price $:$

$\left( \frac{2x+3y}{5} \right) \times \frac{105}{100} = 40$

$\Rightarrow \left( \frac{2x+3y}{5} \right) \times \frac{21}{20} = 40 \quad \longrightarrow (2)$

On equalling equation $(1),$ and $(2),$ we get

$\left( \frac{3x+2y}{5} \right) \times \frac{11}{10} = \left( \frac{2x+3y}{5} \right) \times \frac{21}{20}$

$\Rightarrow (3x+2y) \times 22 = (2x+3y) \times 21$

$\Rightarrow 66x + 44y = 42x + 63y$

$\Rightarrow 24x = 19y$

$\Rightarrow \boxed{\frac{x}{y} = \frac{19}{24}}$

$\therefore$ The cost price per kg of $A$ and $B$ is $19 : 24.$

Correct Answer $: \text{B}$
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