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A wholesaler bought walnuts and peanuts, the price of walnut per kg being thrice that of peanut per kg. He then sold $8$ kg of peanuts at a profit of $10\%$ and $16$ kg of walnuts at a profit of $20\%$ to a shopkeeper. However, the shopkeeper lost $5$ kg of walnuts and $3$ kg of peanuts in transit. He then mixed the remaining nuts and sold the mixture at Rs. $166$ per kg, thus making an overall profit of $25\%$. At what price, in Rs. per kg, did the wholesaler buy the walnuts?

- $98$
- $96$
- $84$
- $86$

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Let the cost price of peanuts per kg be $\text{Rs}. x.$

Then, the cost price of walnuts per kg be $\text{Rs}. 3x.$

A wholesaler sold $8 \; \text{kg}$ of peanuts at a profit of $10\%.$

So, the cost price of $8 \; \text{kg}$ peanuts $ = \text{Rs}. 8x.$

Therefore, selling price of $8 \; \text{kg}$ peanuts $ = 8x \times \frac{110}{100} = \text{Rs.} \frac{88x}{10} $

And, he also sold $16 \; \text{kg}$ walnuts at a profit of $20 \%.$

So, the cost price of $16 \; \text{kg}$ walnuts $ = 16 \times 3x = \text{Rs.} 48x $

Therefore, selling price of $16 \; \text{kg}$ walnuts $ = 48x \times \frac{120}{100} = \text{Rs.} \frac{288x}{5} $

The shopkeeper who bought the products from the wholesaler lost $5 \; \text{kg}$ of walnuts, and $3 \; \text{kg}$ of peanuts in transit.

So, he finally have $11 \; \text{kg}$ of walnuts, and $5 \; \text{kg}$ of peanuts. He then mixed the remaining nuts, and this brought the total quantity $ = (11 \; \text{kg} + 5 \; \text{kg}) = 16 \; \text{kg}.$

Shopkeeper sold $16 \; \text{kg}$ of mixture at the rate of $\text{Rs.} 166 \; \text{per kg}.$

So, total amount earned by shopkeeper of $16 \; \text{kg} = 16 \times 166 = \text{Rs.} 2656 $

Now, since the shopkeeper made a profit of $25 \%$ on his entire purchase of $8 \; \text{kg}$ of peanuts and $16 \; \text{kg}$ of walnuts.

Thus, $125 \% \left( \frac{88x}{10} + \frac{288x}{5} \right) = 2656 $

$ \Rightarrow \frac{125}{100} \left( \frac{88x}{10} + \frac{288x}{5} \right) = 2656 $

$ \Rightarrow \frac{5}{4} \left ( \frac{88x + 576x}{10} \right) = 2656 $

$ \Rightarrow \frac{664x}{8} = 2656 $

$ \Rightarrow 83x = 2656 $

$ \Rightarrow x = \frac{2656}{83} $

$ \Rightarrow \boxed{x = 32} $

$ \therefore$ The cost price of walnuts per kg $ = 3x = 3 \times 32 = \text{Rs.} 96 $

Correct Answer $: \text{B}$

Then, the cost price of walnuts per kg be $\text{Rs}. 3x.$

A wholesaler sold $8 \; \text{kg}$ of peanuts at a profit of $10\%.$

So, the cost price of $8 \; \text{kg}$ peanuts $ = \text{Rs}. 8x.$

Therefore, selling price of $8 \; \text{kg}$ peanuts $ = 8x \times \frac{110}{100} = \text{Rs.} \frac{88x}{10} $

And, he also sold $16 \; \text{kg}$ walnuts at a profit of $20 \%.$

So, the cost price of $16 \; \text{kg}$ walnuts $ = 16 \times 3x = \text{Rs.} 48x $

Therefore, selling price of $16 \; \text{kg}$ walnuts $ = 48x \times \frac{120}{100} = \text{Rs.} \frac{288x}{5} $

The shopkeeper who bought the products from the wholesaler lost $5 \; \text{kg}$ of walnuts, and $3 \; \text{kg}$ of peanuts in transit.

So, he finally have $11 \; \text{kg}$ of walnuts, and $5 \; \text{kg}$ of peanuts. He then mixed the remaining nuts, and this brought the total quantity $ = (11 \; \text{kg} + 5 \; \text{kg}) = 16 \; \text{kg}.$

Shopkeeper sold $16 \; \text{kg}$ of mixture at the rate of $\text{Rs.} 166 \; \text{per kg}.$

So, total amount earned by shopkeeper of $16 \; \text{kg} = 16 \times 166 = \text{Rs.} 2656 $

Now, since the shopkeeper made a profit of $25 \%$ on his entire purchase of $8 \; \text{kg}$ of peanuts and $16 \; \text{kg}$ of walnuts.

Thus, $125 \% \left( \frac{88x}{10} + \frac{288x}{5} \right) = 2656 $

$ \Rightarrow \frac{125}{100} \left( \frac{88x}{10} + \frac{288x}{5} \right) = 2656 $

$ \Rightarrow \frac{5}{4} \left ( \frac{88x + 576x}{10} \right) = 2656 $

$ \Rightarrow \frac{664x}{8} = 2656 $

$ \Rightarrow 83x = 2656 $

$ \Rightarrow x = \frac{2656}{83} $

$ \Rightarrow \boxed{x = 32} $

$ \therefore$ The cost price of walnuts per kg $ = 3x = 3 \times 32 = \text{Rs.} 96 $

Correct Answer $: \text{B}$