AMC 10 · 2003 · #11
Grade 6 rate-ratioPick an answer.
There are two different clocks ticking, so the safe move is to keep the units straight (Tool #8): every quantity is either "watch minutes" or "real minutes", and the one hour of data converts between them. Turn that hour into a unit rate — watch minutes per real minute — and the whole problem becomes a single conversion. Measure both times from noon in minutes so nothing has to be carried across an hour boundary (Tool #4), then write the conversion as one proportion and solve it (Tool #13). Finally convert the answer back into clock form and check it against the choices, since a near-miss choice sits only one minute away (Tool #3).
Turn the observed hour into minutes
In shared units the watch gains 57.6 minutes for every 60 real ones.
Seconds and minutes cannot be compared side by side, so push everything into minutes first.
5.MD.A.1Analyze The UnitsFind the watch-to-real unit rate
That fixes the rate at 24 watch minutes per 25 real ones.
One hour of evidence fixes the exchange rate between watch time and real time, and a fixed rate can be applied to any stretch of time.
One hour of evidence fixes the exchange rate between watch minutes and real minutes.
▸ Why?
The watch runs at a steady rate, so the amount it shows is that rate multiplied by the real time.
▸ Why?
A fixed rate keeps the two times in a constant ratio, so it can be applied to any stretch, not just the measured hour.
Measure the target on the watch
The target is on the watch, so it is 600 watch minutes after noon.
The 10:00 PM in the problem is a watch reading, so it has to be counted in watch minutes before the rate can be used.
4.MD.A.1Introduce A VariableSolve the proportion
Scaling the right way round gives 625 real minutes.
Since 600 watch minutes is exactly 25 blocks of 24, real time is exactly 25 blocks of 25.
6.RP.A.3Convert To AlgebraConvert back to a clock time
That is 10:25 PM, and a minute earlier the watch has not arrived, choice (C).
Converting back and re-running the rate forward closes the loop and rules out the choice one minute earlier.
6.NS.B.3Eliminate PossibilitiesA steadily slow watch runs on its own scale — 24 watch minutes for every 25 real minutes — so multiply whatever the watch says by 25/24 to get the real time.
- Turn the observed hour into minutes
- Find the watch-to-real unit rate
- Measure the target on the watch
- Solve the proportion
- Convert back to a clock time