AMC 10 Step by Step
Log inSign up
Printable

2008 AMC 10B Problem 12

Problem onlyBack to the problem
2008 AMC 10B Problem 12AMC 10 Step by Step
Name:                             Date: September 13, 2026
Sets, Estimation & Miscellaneous · difficulty 2 of 5 · about 2 min · position P11-15

Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on the next step. Pete plans to determine his mileage for a year. On January 1 Pete sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. Pete takes 1800 steps per mile. Which of the following is closest to the number of miles Pete walked during the year?

  1. A)

    25002500

  2. B)

    30003000

  3. C)

    35003500

  4. D)

    40004000

  5. E)

    45004500

Answer

A

Key insight

Each flip is 100,000 steps, so the year totals 44(100,000) + 50,000 = 4,450,000 steps; divided by 1800 that is about 2470 miles.

Solution

Going from 0000000000 back around to 0000000000 takes exactly 100,000100{,}000 steps (the step from 9999999999 to 0000000000 is the hundred-thousandth). Forty-four flips therefore account for 44⋅100,000=4,400,00044 \cdot 100{,}000 = 4{,}400{,}000 steps, and the final reading adds 50,00050{,}000 more:
4,400,000+50,000=4,450,000 steps.4{,}400{,}000 + 50{,}000 = 4{,}450{,}000 \text{ steps}.
Convert to miles:
4,450,0001800=44,50018≈2472.\frac{4{,}450{,}000}{1800} = \frac{44{,}500}{18} \approx 2472.
The nearest choice is 25002500.

The answer is (A) 2500\boxed{\textbf{(A)}\ 2500}.

Why this works

A counter that rolls over is a base-10510^5 odometer: the number of rollovers is the "high digit," and the final reading is the remainder. Once the step count is assembled, the question is a single unit conversion, and "closest to" signals that a rough division (4.454.45 million over 18001800, roughly 4.54.5 million over 1800=25001800 = 2500) is enough.

The trap

Dividing carelessly and landing near 3000, or counting 45 flips; a clean estimate 4,450,000/1800 = 2472 rounds to 2500.

Common mistakes

  • Dividing carelessly and landing near 3000, or counting 45 flips; a clean estimate 4,450,000/1800 = 2472 rounds to 2500.
  • Using 99,99999{,}999 steps per flip; this changes the total by only 4444 steps but wastes time, and can cause an arithmetic slip.

Techniques

Bound the quantity above/below or estimate to pin it down