2012 AMC 10B Problem 13

P11-15~3 minPrint

Linear Equations & Word Problems Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

It 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?

Show solution without answering

Key insight

Speeds add, not times. Pretend the escalator has 120 steps: Clea walks 2 steps per second, and with the escalator she moves 5, so the escalator alone moves 3.

Solution

Pick a length for the escalator that both $60$ and $24$ divide evenly. $120$ works, so pretend the escalator is $120$ steps long.
On the stopped escalator Clea walks $120$ steps in $60$ seconds. Her walking speed is $120 \div 60 = 2$ steps per second.
On the moving escalator she covers $120$ steps in $24$ seconds, which is $120 \div 24 = 5$ steps per second.
That $5$ is her walking speed plus the escalator's speed. So the escalator by itself moves $5 - 2 = 3$ steps per second.
If she just stands, only the escalator moves her: $120 \div 3 = 40$ seconds.

The answer is $\boxed{\textbf{(B)}\ 40}$ .

Why this works

When two motions work together, their speeds add. So turn each time into a speed, do the adding or subtracting with speeds, and turn the answer back into a time at the end. This is the same as "work" problems: if Clea and the escalator together finish the job in $24$ seconds and Clea alone takes $60$ , how long does the escalator take alone?

Alternative approach

Use fractions of the escalator instead. Walking alone, Clea covers $\frac{1}{60}$ of it each second. Walking on the moving escalator, she covers $\frac{1}{24}$ each second. The escalator's share is
$\frac{1}{24} - \frac{1}{60} = \frac{5}{120} - \frac{2}{120} = \frac{3}{120} = \frac{1}{40}$
of the escalator per second, so it takes $40$ seconds alone.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Measure distance in "escalator lengths." Walking on a stopped escalator, Clea covers the whole length in $60$ seconds, so her walking speed is $\frac{1}{60}$ length per second.

Walking on the moving escalator, her speed relative to the ground is her walking speed plus the escalator's speed, and she covers the length in $24$ seconds: combined speed $\frac{1}{24}$ .

Therefore the escalator's own speed is
$\frac{1}{24} - \frac{1}{60} = \frac{5}{120} - \frac{2}{120} = \frac{3}{120} = \frac{1}{40}$
length per second. Standing still, she needs $40$ seconds.

The answer is $\boxed{\textbf{(B)}\ 40}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 13: Data and Statistics (§13.1)