2008 AMC 10B Problem 12

P11-15~2 minPrint

Sets, Estimation & Miscellaneous Ratios, Percents & Averages Bound the quantity above/below or estimate to pin it down

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?

Show solution without answering

Key insight

Each flip means 100,000 steps. Forty-four flips plus the final 50,000 give 4,450,000 steps, and dividing by 1800 gives about 2470 miles.

Solution

First find the total number of steps, then turn steps into miles.

1. Going from $00000$ all the way around back to $00000$ takes $100{,}000$ steps: $99{,}999$ steps to reach $99999$ , and one more step to flip.
2. There were $44$ flips, so they account for
$44 \times 100{,}000 = 4{,}400{,}000 \text{ steps}.$
3. After the last flip the counter climbed to $50000$ , which adds $50{,}000$ steps. The total is
$4{,}400{,}000 + 50{,}000 = 4{,}450{,}000 \text{ steps}.$
4. Each mile is $1800$ steps, so the miles are
$\frac{4{,}450{,}000}{1800} = \frac{44{,}500}{18} \approx 2472.$
5. The closest choice is $2500$ .

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

Why this works

A counter that flips over works like a car's odometer: the number of flips counts the big chunks ( $100{,}000$ steps each), and the final reading is what is left over. The words "closest to" tell you a rough division is enough. For example, $4{,}500{,}000 \div 1800 = 2500$ already points to (A).

Concise solution

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

Going from $00000$ back around to $00000$ takes exactly $100{,}000$ steps (the step from $99999$ to $00000$ is the hundred-thousandth). Forty-four flips therefore account for $44 \cdot 100{,}000 = 4{,}400{,}000$ steps, and the final reading adds $50{,}000$ more:
$4{,}400{,}000 + 50{,}000 = 4{,}450{,}000 \text{ steps}.$
Convert to miles:
$\frac{4{,}450{,}000}{1800} = \frac{44{,}500}{18} \approx 2472.$
The nearest choice is $2500$ .

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

Review in the AoPS books

Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.2, §2.3)
Introduction to Algebra, Chapter 6: Ratios and Percents
Prealgebra, Chapter 7: Ratios, Conversions, and Rates (§7.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)