Question

A base- $10$ three-digit number $n$ is selected at random. Which of the following is closest to the probability that the base- $9$ representation and the base- $11$ representation of $n$ are both three-digit numerals?

$0.3$

$0.4$

$0.5$

$0.6$

$0.7$

Answer: E

Concepts:number base basic probability counting integers in a range

Difficulty rating: 1660

Small Hint:

A number is three-digit in base $9$ when $81 \le n \le 728,$ and three-digit in base $11$ when $121 \le n \le 1330$

Big Hint:

Among the $900$ three-digit base- $10$ numbers, count those with $121 \le n \le 728$

Solution:

The largest three-digit base- $9$ number is $9^3 - 1 = 728,$ and the smallest three-digit base- $11$ number is $11^2 = 121.$

So both conditions hold exactly when $121 \le n \le 728,$ giving $608$ integers.

Out of $900$ three-digit numbers, the probability is $\dfrac{608}{900} \approx 0.7.$

Thus, the correct answer is E.

2003 AMC 10A Problem 20

Answer: E

Difficulty rating: 1660

Small Hint:

Big Hint:

Solution:

Problem 20 in Other Years