AMC 10 · 2012 · #13

Grade 6 arithmetic
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Constant-Rate Closing Speed
ratefraction-arithmetic convert-to-algebra ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
With the escalator switched off, Clea walks all the way down it in 60 seconds. With the escalator running, walking down it the same way takes her 24 seconds. Find how many seconds the trip takes if she stands still on the running escalator and lets it carry her down.

Pick an answer.

(A)
36
(B)
40
(C)
42
(D)
48
(E)
52

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

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.

1STEP 1

Pin down what moving adds

Both trips share one number: Clea's speed w along the steps. In t seconds she descends wt from her legs plus et from the moving belt.

distance down the shaft in t seconds = wt_her legs + et_the steps
2STEP 2

Turn each trip into a rate

Make one full ride the unit, so the unknown length cancels: w = 1/60 escalator per second, and w + e = 1/24.

w = 1/60, w + e = 1/24 (escalators per second)
3STEP 3

Subtract to isolate the escalator

Take her walking out of the combined rate: e = 1/24 - 1/60 = 5/120 - 2/120 = 1/40 escalator per second.

e = 1/24 - 1/60 = 5/120 - 2/120 = 3/120 = 1/40
4STEP 4

Turn the rate back into time

Standing still leaves only e, so one whole escalator takes 1 ÷ 1/40 = 40 seconds → (B).

t = 1/e = 1 ÷ 1/40 = 40 seconds → (B)
Answer
40
Rebuild the given trip from the answer: 1/60 + 1/40 = 2/120 + 3/120 = 5/120 = 1/24, which is exactly the 24-second walking-while-running trip, so 40 reproduces both givens rather than just one. It also passes the crude bound: removing her walking can only slow the trip, so the ride alone must take more than 24 seconds, and 40 ＞ 24. The two traps sit right beside the answer. Subtracting the times gives 60 - 24 = 36, choice (A) — but times do not add, and testing it fails: with the escalator at 1/36, the joint trip would take 1 ÷ (1/60 + 1/36) = 22.5 seconds, not 24. Averaging the times gives (60 + 24)/2 = 42, choice (C), which has no justification at all.
💡Key takeaway

Times 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

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