26 steps of an escalator she needs 30 seconds to get to the bottom. But if on another trip she makes 34 steps she only needs 18

Question

Harriet finds if she walks down 26 steps of an escalator she needs 30 seconds to get to the bottom. But if on another trip she makes 34 steps she only needs 18 seconds to reach the bottom.

If the time measured from the instant that the too step begins to descend until the time Harriet steps off the last step at he bottom onto the level platform, what is the height of the stairway in steps?

How many seconds does it take the escalators to travel one step?

Answer 1

First premise is that walking on the escalator of course doesn't change its rate of motion.  The problem states: with 8 steps fewer on second trip (34 - 16) she saves 12 seconds (30 - 18).  So, the rate of the escalator is 12/8 secs per step (or 1.5 sec per step).  To determine the total number of escalator steps, think of Harriet riding fully without taking any steps toward bottom, or skipping all the steps (instantly). The key point to understand is that a linear relationship exists between number of steps and time to complete a trip.

 

I'll show two methods of solving...

Tabulating:

# steps taken     trip time

26                      30 (given)

34                      18*

35                      16.5 (working step by step...)

36                      15

37                      13.5

38                      12

39                      10.5

40                      9

41                      7.5

42                       6

43                      4.5

44                        3

45                      1.5 

46                      0    (final step is presumably where Harriet hops off at bottom)

 

or mathematically: *  knowing 18 seconds needs to be equated to #steps, the conversion is 18/1.5=12 steps (more than 34 as in the table above).

34+12=46 steps total

 

Double check:   46 steps * 1.5 sec/step = 69 secs to ride full length 

  Yes, it seems correct that if there are 46 steps and Harriet "skips" 26 of them (about 56%), she'll save that percentage of the total time..  69 secs * (1 - 0.56) = 30 secs

 

Like?