AMC 10 · 2012 · #13
Grade 6 arithmeticIt takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Clea can walk down an escalator in $60$ seconds when it is switched off, and in $24$ seconds when it is moving and she walks. Find how long the ride takes if she just stands still on the moving escalator.
Givens: Walking down the stopped escalator takes $60$ seconds; Walking down the moving escalator takes $24$ seconds; The escalator is the same length in every case; Answer choices: (A) $36$, (B) $40$, (C) $42$, (D) $48$, (E) $52$
Unknowns: The number of seconds to ride the moving escalator while standing still
Understand
Restated: Clea can walk down an escalator in $60$ seconds when it is switched off, and in $24$ seconds when it is moving and she walks. Find how long the ride takes if she just stands still on the moving escalator.
Givens: Walking down the stopped escalator takes $60$ seconds; Walking down the moving escalator takes $24$ seconds; The escalator is the same length in every case; Answer choices: (A) $36$, (B) $40$, (C) $42$, (D) $48$, (E) $52$
Plan
Primary tool: #8 Analyze the Units
Secondary: #9 Solve an Easier Related Problem, #4 Introduce a Variable
This is a rate problem, so Tool #8 (Analyze the Units) keeps every quantity as a speed in "units per second," which makes the key fact usable: walking speed and escalator speed add up. Tool #9 (Solve an Easier Related Problem) lets us fix the escalator length at a convenient number so the rates become plain whole numbers. Tool #4 (Introduce a Variable) is the lens for naming the two hidden speeds and subtracting to isolate the escalator's own speed.
Execute — Answer: B
6.RP.A.2 Step 1 Pick a convenient length
- The length never changes, so we may choose any value for it.
- Pick $120$ units, the least common multiple of $60$ and $24$, so both times divide it evenly.
💡 A concrete length turns each time into a clean whole-number speed.
6.RP.A.2 Step 2 Clea's walking speed
- On the stopped escalator she covers all $120$ units in $60$ seconds by walking.
- Her walking speed is $120 \div 60 = 2$ units per second.
💡 Speed is just distance shared out evenly over the time it took.
6.RP.A.2 Step 3 Combined walking-plus-escalator speed
- On the moving escalator she walks and the stairs move too, covering $120$ units in $24$ seconds.
- Their combined speed is $120 \div 24 = 5$ units per second.
💡 Walking on a moving escalator, the two speeds push together in the same direction.
6.RP.A.3 Step 4 Isolate the escalator's speed
- The combined speed is Clea's walking speed plus the escalator's speed.
- Subtracting her $2$ units per second leaves the escalator's own speed: $5 - 2 = 3$ units per second.
💡 Take away the part you know (her walking) to reveal the part you want (the escalator).
6.RP.A.3 Step 5 Time to just stand
- Standing still, only the escalator carries her, at $3$ units per second over $120$ units.
- The time is $120 \div 3 = 40$ seconds, so the answer is (B).
💡 Time is the distance divided by whatever single speed is moving her.
6.RP.A.2 The length never changes, so we may choose any value for it. Pick $120$ units, t 6.RP.A.2 On the stopped escalator she covers all $120$ units in $60$ seconds by walking. 6.RP.A.2 On the moving escalator she walks and the stairs move too, covering $120$ units 6.RP.A.3 The combined speed is Clea's walking speed plus the escalator's speed. Subtracti 6.RP.A.3 Standing still, only the escalator carries her, at $3$ units per second over $12 Review
Reasonableness: The standing time $40$ seconds sits between $24$ (walking a moving escalator, the fastest case) and $60$ (walking a dead escalator), which is exactly where riding alone should land — slower than walking-plus-riding but faster than walking on a still escalator. It is also consistent with rates: $24$s (walk+ride) and $60$s (walk only) give speeds $5$ and $2$, whose difference $3$ predicts $120/3 = 40$s. Choices like (A) $36$ or (C) $42$ do not come from any clean rate subtraction.
Alternative: Use fractions of the escalator per second instead of a fixed length. Walking clears $\tfrac{1}{60}$ of it each second; walking while riding clears $\tfrac{1}{24}$ each second. The escalator alone clears $\tfrac{1}{24} - \tfrac{1}{60} = \tfrac{5-2}{120} = \tfrac{3}{120} = \tfrac{1}{40}$ per second, so a full ride takes $40$ seconds — matching (B).
CCSS standards used (min grade 6)
6.RP.A.2Understand the concept of a unit rate and use rate language (Converting each time into a speed in units per second: $120/60 = 2$ and $120/24 = 5$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Subtracting speeds to isolate the escalator ($5 - 2 = 3$) and dividing to get the standing time ($120/3 = 40$).)
⭐ Turn each time into a speed, and since walking and the escalator add up, subtract your walking speed to see how fast the escalator alone carries you.
⭐ Turn each time into a speed, and since walking and the escalator add up, subtract your walking speed to see how fast the escalator alone carries you.
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