Two stores A and B mark the price of an item identically. A allows $3$ successive discounts of $10\%$ each. B allows $10\%$ discount on the list price and a subsequent discount of $19\%$. Under the circumstances, which of the following is true?

### Concept & Logic To compare different discount structures, find the final multiplier or equivalent discount for each store. $$ \text{Final Price} = \text{List Price} \times \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \dots $$ ### Step-by-Step Solution * **Given:** * Store A discounts = $10\%$, $10\%$, $10\%$ * Store B discounts = $10\%$, $19\%$ * **Calculation:** * Assume the marked price is $100$. * **Store A:** Price after 1st discount = $100 \times 0.9 = 90$. Price after 2nd discount = $90 \times 0.9 = 81$. Price after 3rd discount = $81 \times 0.9 = 72.9$. * **Store B:** Price after 1st discount = $100 \times 0.9 = 90$. Price after 2nd discount = $90 \times (1 - 0.19) = 90 \times 0.81$. $90 \times 0.81 = 72.9$. * Since both final prices are $72.9$, the price is the same at both stores. ### Exam Strategy & Shortcut Notice that Store A gives three $10\%$ discounts. The first two combine to $10 + 10 - 1 = 19\%$. So, Store A effectively gives successive discounts of $19\%$ and $10\%$. Store B gives successive discounts of $10\%$ and $19\%$. Since multiplication is commutative, a $19\%$ discount followed by a $10\%$ discount is exactly the same as a $10\%$ discount followed by a $19\%$ discount. Thus, they must be equal. ### Common Pitfall Trying to estimate the answer by adding the percentages (Store A: $30\%$, Store B: $29\%$) and incorrectly concluding that A is cheaper. ### Final Answer Therefore, the correct answer is **The price of the article is same at A and B.**