A shopkeeper marks the price of an item $50\%$ above its cost. During a festival, he offers a sequence of discounts — first a $20\%$ discount, then an additional $10\%$ off on the reduced price, and finally gives a further rebate of $₹30$ on the total bill for 10 identical items. If after all these reductions, he still makes a profit of $5\%$ on selling the 10 items, what is the cost price of one item?

Let the Cost Price (CP) of one item be $x$.

Since there are 10 identical items:

Total Cost Price $= 10x$

The Marked Price (MP) is $50\%$ above the CP.

MP of one item $= x + 50\%$ of $x$

$= 1.5x$

Total Marked Price (for 10 items) $= 10 \times 1.5x$

$= 15x$

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Applying the first discount of $20\%$:

Price after first discount $= 15x \times (1 - \frac{20}{100})$

$= 15x \times 0.8$

$= 12x$

Applying the second discount of $10\%$ on the reduced price:

Price after second discount $= 12x \times (1 - \frac{10}{100})$

$= 12x \times 0.9$

$= 10.8x$

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After the rebate of $₹30$ on the total bill:

Final Selling Price (SP) $= 10.8x - 30$

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The shopkeeper makes a total profit of $5\%$ on the Total Cost Price ($10x$).

Final SP $= \text{Total CP} + \text{Profit}$

$= 10x + (5\%$ of $10x)$

$= 10x + 0.5x$

$= 10.5x$

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Equating the two expressions for Final Selling Price:

$10.8x - 30 = 10.5x$

$10.8x - 10.5x = 30$

$0.3x = 30$

$x = \frac{30}{0.3}$

$x = 100$

Therefore, the cost price of one item is $₹100$.