AMC 10 · 2008 · #12

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

place-valueunit-conversionestimation identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 2 insights

Problem

Postman Pete's pedometer counts up to 99999 and then rolls over to 00000 on the next step. He sets it to 00000 on January 1. During the year the pedometer rolled over 44 times, and on December 31 it reads 50000. Pete walks 1800 steps per mile. Which answer choice is closest to the number of miles Pete walked during the year?

Pick an answer.

(A)

$\ 2500$

(B)

$\ 3000$

(C)

$\ 3500$

(D)

$\ 4000$

(E)

$\ 4500$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The chain of units is the spine of the problem: the pedometer measures steps, but the answer is in miles, and 1800 steps per mile is the rate that bridges them, so tool #8 (Analyze the Units) leads. Getting the step count first requires tool #7 (Identify Subproblems): one subproblem is how many steps a single rollover represents, another is adding the leftover 50000. Once the estimate comes out near 2472, tool #3 (Eliminate Possibilities) finishes it, since only one choice is close to that value.

1STEP 1

What one rollover means

Five slots stop at 99999, so the next step shows 00000 — yet that step happened. One rollover is 100000 real steps.

99999 + 1 = 100000

2STEP 2

Total steps for the year

The 44 cycles give 44 × 100000 = 4400000 steps; add the 50000 still on the dial to reach 4450000 steps.

44 × 100000 + 50000 = 4400000 + 50000 = 4450000

3STEP 3

Set up the step-to-mile conversion

Steps and miles are linked by 1800 steps per mile, so dividing the step total by 1800 cancels steps and leaves miles.

miles = (4450000 steps)/(1800 steps/mile)

4STEP 4

Divide and pick the closest choice

4450000 ÷ 1800 ≈ 2472 miles — only 28 from 2500, but over 500 from every other choice, so (A) wins.

4450000 ÷ 1800 ≈ 2472.2 → closest to 2500

Answer

2500

Sanity-check the size: 44 rollovers of 100000 is 4.4 million steps, and at roughly 2000 steps per mile that is a little over 2000 miles, so an answer in the low thousands is expected — 2472 fits, and 3000 or more would be too big. Using the slightly larger true rate of 1800 (fewer steps per mile than 2000 means more miles) nudges it up to about 2472, still closest to 2500. The estimate lands well below the other choices, confirming (A).

💡Key takeaway

A rollover counter starts over every 100000 steps, so add up the full cycles plus the leftover, then divide by the steps in one mile.

What one rollover means
Total steps for the year
Set up the step-to-mile conversion
Divide and pick the closest choice

What one rollover means

Total steps for the year

Set up the step-to-mile conversion

Divide and pick the closest choice

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