AMC 10 · 2018 · #3
Grade 6 arithmeticPick an answer.
AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The lifetime is a count of seconds, but the answer is a date. The whole problem is a unit-conversion chain: seconds → minutes → hours → days. Analyzing the units tells you exactly which numbers to divide by. Splitting the work into subproblems (compute 10!, convert to days, count forward on the calendar) keeps each piece simple, and a calendar count finishes the job.
Compute the number of seconds
Multiply the chain 10·9·8·⋯·1 one factor at a time to get 3,628,800 seconds.
A factorial is just a long multiplication problem, so do it one factor at a time.
5.NBT.B.5Identify SubproblemsSet up the unit chain
Seconds aren't a calendar unit, so divide by 60, then 60, then 24 to step seconds → minutes → hours → days.
Going to a larger unit means dividing by how many small ones fit inside it.
5.MD.A.1Analyze The UnitsCarry out the divisions
Running the divisions gives 60,480 minutes, then 1,008 hours, and finally 42 days.
Dividing in stages keeps each step a clean whole number instead of one huge division.
6.NS.B.2Analyze The UnitsCount 42 days on the calendar
From noon January 1, 31 days reaches February 1; the remaining 42−31=11 days land on February 12.
Counting days is easiest if you jump to the end of the month first, then count the leftover days.
4.MD.A.2Draw A DiagramTurn the seconds into days by dividing, then just count the days forward on the calendar.
- Compute the number of seconds
- Set up the unit chain
- Carry out the divisions
- Count 42 days on the calendar