AMC 10 · 2009 · #10
Grade 5 countingPick an answer.
The display has two independent parts: the hour and the minute. A time is correct only when the hour is correct AND the minute is correct, and the set of correct minutes is the same in every hour. So split the problem: find the fraction of hours with no 1, find the fraction of minutes with no 1, then multiply the two fractions.
Reframe: correct means no digit is 1
Correct means that digit never appears.
If the machine only ever mangles the digit 1, then avoiding every 1 guarantees a perfect display.
4.NBT.A.2Change Focus Count The ComplementCount the correct hours
Counting the hours gives 2/3.
Just cross out every hour that has a 1 somewhere and count what survives.
4.NF.A.1Make A Systematic ListCount the correct minutes
Counting the minutes gives 3/4.
Every allowed tens digit can sit next to every allowed ones digit, so multiply the choices.
Every allowed tens digit can sit next to every allowed ones digit, so the choices multiply.
▸ Why?
The two digit places are chosen without regard to each other, so their counts multiply.
▸ Why?
Every minute of the hour is just as likely as any other, so counting them gives the chance directly.
Combine the two subproblems
The two are independent, so multiplying gives 1/2, choice (E).
Correct hour and correct minute must both happen, so the fractions multiply.
5.NF.B.4Identify SubproblemsSplit a two-part display into its parts, find the fraction right in each part, then multiply the fractions to get the fraction right overall.
- Reframe: correct means no digit is 1
- Count the correct hours
- Count the correct minutes
- Combine the two subproblems