Study the following information carefully to answer these questions: A young girl Roopa leaves home with $x$ flowers and goes to the bank of a nearby river. On the bank of the river, there are four places of worship, standing in a row. She dips all the $x$ flowers into the river, the number of flowers doubles. Then, she enters the first place of worship and offers $y$ flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the second place of worship and offers $y$ flowers to the deity. She dips the remaining flowers into the river and again the number of flowers doubles. She goes to the third place of worship and offers $y$ flowers to the deity. She dips the remaining flowers into the river and again the number of flowers doubles. She goes to the fourth place of worship and offers $y$ flowers to the deity. Now she is left with no flowers in hand. If Roopa leaves home with 30 flowers, the number of flowers she offers to each deity is

### Concept & Logic This problem is a classic iterative sequence puzzle that translates into a linear equation. The core logic involves tracking the number of flowers through repeated operations of multiplying by 2 (doubling) and subtracting a constant $y$ (offering). By establishing an equation for the final state (zero flowers remaining), we can find the exact relationship between the initial number of flowers $x$ and the offered amount $y$. ### Step-by-Step Solution * **Given:** * Initial flowers = $x = 30$ * Flowers offered at each temple = $y$ * The number of flowers doubles before each offering. * There are 4 temples, and she ends with 0 flowers. * **Calculation:** 1. **After 1st river & temple:** She dips $x$ flowers $\rightarrow$ becomes $2x$. She offers $y$ flowers $\rightarrow$ remaining is $2x - y$. 2. **After 2nd river & temple:** Dips remaining $\rightarrow$ becomes $2(2x - y) = 4x - 2y$. Offers $y \rightarrow$ remaining is $4x - 2y - y = 4x - 3y$. 3. **After 3rd river & temple:** Dips remaining $\rightarrow$ becomes $2(4x - 3y) = 8x - 6y$. Offers $y \rightarrow$ remaining is $8x - 6y - y = 8x - 7y$. 4. **After 4th river & temple:** Dips remaining $\rightarrow$ becomes $2(8x - 7y) = 16x - 14y$. Offers $y \rightarrow$ remaining is $16x - 14y - y = 16x - 15y$. 5. **Equate to final state:** We know she is left with 0 flowers. $$16x - 15y = 0$$ $$16x = 15y$$ 6. **Solve for $y$ given $x = 30$:** $$16(30) = 15y$$ $$480 = 15y$$ $$y = 32$$ ### Exam Strategy & Shortcut Instead of writing out all 4 steps, you can use a generalized formula for this specific type of puzzle. If an initial quantity $x$ doubles $n$ times, and a constant amount $y$ is subtracted after each doubling, the final amount is given by the formula: $$Final = 2^n(x) - (2^n - 1)y$$ For $n = 4$ temples: $$Final = 2^4(x) - (2^4 - 1)y$$ $$0 = 16x - 15y$$ Substitute $x = 30$, and you immediately get $16(30) = 15y \implies y = 32$. ### Common Pitfall A very common mistake is trying to calculate the numbers sequentially with actual values instead of algebra, getting lost in the arithmetic, or forgetting to distribute the factor of 2 when doubling the remaining amounts (e.g., writing $2x - y \rightarrow 4x - y$ instead of the correct $4x - 2y$). Always establish the algebraic relationship first. ### Final Answer Therefore, the correct answer is 32.