AMC 10 · 2012 · #9
Grade 6 rate-ratioPick an answer.
The length of the escalator is never stated, so Tool #8 (Analyze the Units) sets the right currency: measure everything in escalators per second, and the unknown length cancels out on its own. Tool #4 (Introduce a Variable) is needed before any arithmetic, because the problem only works once Clea's walking speed is named as a speed relative to the steps — that is what makes the same number appear in both trips. Tool #16 (Change Focus / Count the Complement) then flips attention from Clea to the escalator: the steps supply whatever share of the trip her legs did not. Tool #11 (Work Backwards) turns the escalator's rate back into the time the question actually asks for. The reason to be careful is that the tempting shortcut 60-24=36 subtracts times, and times are not what add here.
Pin down what moving adds
Walking and the stairway each add their own speed.
Her legs and the belt each move her, and the two contributions simply add.
6.RP.A.3Introduce A VariableTurn each trip into a rate
Each trip becomes one rate.
A trip that takes 60 seconds means 1/60 of the trip is finished each second.
A trip that takes sixty seconds means one sixtieth of the trip is finished each second.
▸ Why?
At a steady pace the amount covered is the rate multiplied by the time, so the rate is the reciprocal of the time.
▸ Why?
Her legs and the belt each move her, so the combined rate is the two rates added or subtracted.
Subtract to isolate the escalator
Subtracting rates isolates the stairway.
Rates subtract cleanly even though the times themselves do not.
5.NF.A.1Change Focus Count The ComplementTurn the rate back into time
Flipping it back gives 40, choice (B).
If 1/40 of the ride happens every second, the whole ride takes 40 seconds.
6.NS.A.1Work BackwardsTimes never add, but rates do: 1/24 - 1/60 = 1/40, so standing still takes 40 seconds.
- Pin down what moving adds
- Turn each trip into a rate
- Subtract to isolate the escalator
- Turn the rate back into time