2013 AMC 10B Problem 4

P1-10~2 minPrint

Number Properties Basic Counting Set up the equation/formula and compute; no special trick needed

When counting from $3$ to $201$ , $53$ is the $51^\mathrm{st}$ number counted. When counting backwards from $201$ to $3$ , $53$ is the $n^\mathrm{th}$ number counted. What is $n$ ?

Show solution without answering

Key insight

Counting down from 201 to 53 counts both end numbers, so the count is 201 minus 53, plus 1.

Solution

1. First see the rule on a tiny example. Count down from $10$ to $7$ : you say $10, 9, 8, 7$ . That is $4$ numbers, but $10 - 7 = 3$ . You need one more than the difference, because both ends get said.
2. Test the rule on the fact given in the problem. Counting up from $3$ , the number $53$ is said $53 - 3 + 1 = 51$ st. That matches.
3. Now count down from $201$ . The numbers said, up to and including $53$ , are $201, 200, 199, \ldots, 54, 53$ . How many is that?
$201 - 53 + 1 = 149.$
So $53$ is the $149$ th number said.

The answer is $\boxed{\textbf{(D)}\ 149}$ .

Why this works

To count the whole numbers from $a$ to $b$ , including both ends, use $b - a + 1$ , not $b - a$ . When unsure, check the rule on a list short enough to count by hand; the problem even gave you one such check.

Alternative approach

Count the whole list first: from $3$ to $201$ there are $201 - 3 + 1 = 199$ numbers. Going forward, $50$ numbers come before $53$ , so going backward, the numbers after $53$ are those $50$ . The numbers before and including $53$ in the backward count are $199 - 50 = 149$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Counting backwards from $201$ , the numbers said before reaching $53$ are $201, 200, \dots, 54$ , and then $53$ itself. The position of $53$ is the number of integers from $53$ to $201$ inclusive:
$201 - 53 + 1 = 149.$

The answer is $\boxed{\textbf{(D)}\ 149}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics
Introduction to Counting & Probability, Chapter 1: Counting Is Arithmetic
Introduction to Algebra, Chapter 20: Special Functions (§20.3)
Introduction to Counting & Probability, Chapter 2: Basic Counting Techniques