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

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