AMC 10 · 2008 · #6
Grade 6 arithmeticPick an answer.
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.
What one rollover means
One rollover is the top plus one.
A five-digit counter starts over every 100000 steps, the same way a car odometer clicks back to zeros after a full cycle.
A five-digit counter starts over every hundred thousand steps, the way an odometer clicks back to zeros.
▸ Why?
Once the display fills up it returns to its start, and everything after that repeats the same run.
▸ Why?
Five digit places hold exactly one hundred thousand different readings, so that is the length of one full cycle.
Total steps for the year
Adding the rollovers and the reading gives the total steps.
Total steps are the completed cycles plus whatever is left showing on the dial.
4.OA.A.3Identify SubproblemsSet up the step-to-mile conversion
One conversion turns steps into distance.
Dividing total steps by steps-per-mile cancels the steps and leaves miles.
6.RP.A.3Analyze The UnitsDivide and pick the closest choice
Dividing lands nearest 2500, choice (A).
Once you have the true value 2472, the winning choice is just the one it sits nearest to.
6.NS.B.2Eliminate PossibilitiesA 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