A store raised the price of an item by exactly 10 percent. Which of the following could not be the resulting price of the item?

### Concept & Logic When an item's price is raised by 10%, the new price is 110% of the original price. This means the new price is the original price multiplied by 1.1 (or $\frac{11}{10}$). Assuming the original price was a standard currency amount (like exact cents or rupees without infinite repeating decimals), the new price, when expressed in the smallest currency unit (cents/paise), must be perfectly divisible by 11. $$ \text{New Price} = \text{Old Price} \times \frac{11}{10} $$ ### Step-by-Step Solution * Let the original price in the smallest unit (paise) be $x$. * The new price is $x \times \frac{11}{10}$. * For the original price $x$ to be an integer (a valid exact currency amount), the new price multiplied by 10 must be divisible by 11. Effectively, the new price without the decimal point should be a multiple of 11. * Let's convert the options to integer values (paise) and check for divisibility by 11: * (a) ₹ 5.50 $\rightarrow 550$. $550 / 11 = 50$. (Original price was ₹ 5.00. Valid.) * (b) ₹ 7.60 $\rightarrow 760$. $760 / 11 = 69.09...$ (Original price would be a non-terminating fraction. Invalid.) * (c) ₹ 11.00 $\rightarrow 1100$. $1100 / 11 = 100$. (Original price was ₹ 10.00. Valid.) * (d) ₹ 12.10 $\rightarrow 1210$. $1210 / 11 = 110$. (Original price was ₹ 11.00. Valid.) * Since 7.60 does not yield a clean original price, it cannot be the resulting price. ### Exam Strategy & Shortcut Use the divisibility rule for 11! Discard the decimal point and check the digits. The alternating sum of digits must be 0 or a multiple of 11. (a) $550 \rightarrow 5 - 5 + 0 = 0$ (Divisible) (b) $760 \rightarrow 7 - 6 + 0 = 1$ (Not Divisible) $\leftarrow$ This is the answer! (c) $1100 \rightarrow 1 - 1 + 0 - 0 = 0$ (Divisible) (d) $1210 \rightarrow 1 - 2 + 1 - 0 = 0$ (Divisible) ### Common Pitfall Attempting to find 10% of the *options* and subtracting it. The 10% increase was applied to the *original* unknown price, not the final price. Calculating 10% of 5.50 gives 0.55, which is an incorrect reverse calculation method. You must divide by 1.1, not multiply by 0.9. ### Final Answer **Therefore, the correct answer is ₹ 7.60.**