Anil buys $12$ toys and labels each with the same selling price. He sells $8$ toys initially at $20 \%$ discount on the labeled price. Then he sells the remaining $4$ toys at an additional $25 \%$ discount on the discounted price. Thus, he gets a total of Rs $2112,$ and makes a $10\%$ profit. With no discounts, his percentage of profit would have been

Let's break down this step-by-step to understand how discounts affect profit calculations.

What we know:

Anil buys 12 toys and labels each with the same selling price

8 toys sold at 20% discount on labeled price

4 toys sold at additional 25% discount on the already discounted price

Total revenue = Rs 2112

He makes 10% profit with these discounts

What we need to find: Profit percentage if no discounts were given

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Let's use:

$y$ = labeled price of one toy

$x$ = total cost price of all 12 toys

This makes our calculations cleaner and helps us track what we're solving for.

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Revenue from 8 toys (20% discount):

Discount = 20% of $y$ = $0.20y$

Selling price per toy = $y - 0.20y = 0.80y$

Revenue from 8 toys = $8 \times 0.80y = 6.4y$

Revenue from 4 toys (additional 25% discount):

First, apply 20% discount: $0.80y$ (same as above)

Then, apply additional 25% discount on this discounted price

Additional discount = 25% of $0.80y$ = $0.25 \times 0.80y = 0.20y$

Final selling price per toy = $0.80y - 0.20y = 0.60y$

Revenue from 4 toys = $4 \times 0.60y = 2.4y$

Key insight: "Additional 25% discount on discounted price" means we apply 25% on the already reduced price of $0.80y$, not on the original price $y$.

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Total revenue equation:

$6.4y + 2.4y = 2112$

$8.8y = 2112$

$y = 2112 \div 8.8 = \text{Rs } 240$

Check: $8.8 \times 240 = 2112$

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Since he makes 10% profit:

Profit formula: Selling Price = Cost Price $\times$ (1 + Profit%)

$2112 = x \times (1 + 0.10)$

$2112 = 1.10x$

$x = 2112 \div 1.10 = \text{Rs } 1920$

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Revenue without any discounts:

12 toys $\times$ Rs 240 = Rs 2880

Profit percentage formula:

Profit% = $\dfrac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100\%$

Profit% = $\dfrac{2880 - 1920}{1920} \times 100\%$

Profit% = $\dfrac{960}{1920} \times 100\%$

Profit% = $0.5 \times 100\% = 50\%$

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Therefore, with no discounts, his percentage of profit would have been 50%.