Question

A store owner decided to raise the price of a particular item by exactly 10%. Of the following which is NOT the new price?

• A. $1.10
• B. $8.80
• C. $11.00
• D. $57.30
• E. $78.10

Hint:

A number is divisible by 1.1 if it is divisible by 11 after moving the decimal point. For example, to check 57.30, we check if 573 is divisible by 11. The rule for divisibility by 11 is to check if the alternating sum of the digits is a multiple of 11. For 573, the sum is 3 - 7 + 5 = 1, which is not a multiple of 11. Thus, 573 is not divisible by 11.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When a price is increased by 10%, the new price becomes 110% of the original price.
That means:
\[ N = 1.1 \times P \] where \(N\) is the new price and \(P\) is the original price.
Thus, to check validity, compute: \[ P = \frac{N}{1.1} \] If \(P\) is a sensible monetary value (up to two decimals), then \(N\) is valid.

Step 2: Testing Each Option:
(A) For $1.10:
\[ P = \frac{1.10}{1.1} = 1.00 \] Valid.

(B) For $8.80:
\[ P = \frac{8.80}{1.1} = 8.00 \] Valid.

(C) For $11.00:
\[ P = \frac{11.00}{1.1} = 10.00 \] Valid.

(D) For $57.30:
\[ P = \frac{57.30}{1.1} = \frac{573}{11} \approx 52.090909... \] This is not a terminating decimal with two places, so not a valid price.

(E) For $78.10:
\[ P = \frac{78.10}{1.1} = 71.00 \] Valid.

Step 3: Final Answer:
The price $57.30 is the only one that cannot be obtained from a 10% increase on a valid original price.