A girl wants to count the steps of a moving escalator which is going up. If she is going up on it, she counts 60 steps. If she is walking down, taking the same time per step, then she counts 90 steps. How many steps would she have to take in either direction, if the escalator were standing still?

Let's V the speed of the girl in step
v the speed of the escalator in step
L the length between ground and floor.

V*60+v*60=L ==> V+v=L/60
V*90-v*90=L ==> V-v=L/90
2V=L(3+2)/180
==>V=5L/360
Time to climb (in step) with v=0 (escalator standing still)
V*t=L==> t=L/V=L/(L*5/360)=72

The girl will count 72 steps.

Answer Link

AkshayG AkshayG · Answer 2 · 2019-11-03T08:46:54+01:00

The number of steps she have to take in either direction is [tex]\boxed{72{\text{ steps}}}.[/tex]

Further explanation:

The formula of the speed can be expressed as follows,

[tex]\boxed{{\text{Speed}} = \frac{{{\text{Distance}}}}{{{\text{Time}}}}}[/tex]

The formula of the time can be expressed as follows,

[tex]\boxed{{\text{Time}} = \frac{{{\text{Distance}}}}{{{\text{Speed}}}}}[/tex]

Given:

If a girl is going up then the numbers of steps she counts are 60.

If a girl is going down then the numbers of steps she counts are 90.

Explanation:

Consider the speed of the girls be “x”.

Consider the speed of the escalator be “y”.

Consider the distance between the ground and the floor will be “d”.

If a girl is going upward then the speed will be [tex]x - y[/tex]

[tex]x - y = \dfrac{d}{{60}}[/tex] …… (1)

If a girl is going downward then the speed will be [tex]x + y[/tex]

[tex]x + y = \dfrac{d}{{90}}[/tex] …… (2)

Now add both the equations.

[tex]\begin{aligned}x - y + x + y &= \frac{d}{{60}} + \frac{d}{{90}}\\2x &= \frac{{90d + 60d}}{{60 \times 90}}\\2x &= \frac{{150d}}{{5400}}\\2x &= \frac{{5d}}{{180}}\\x &= \frac{{5d}}{{360}}\\\end{aligned}[/tex]

The time in steps can be calculated as follows,

[tex]\begin{aligned}t&=\dfrac{d}{{\dfrac{{5d}}{{360}}}}\\&= \dfrac{{360}}{5}\\&= 72\\\end{aligned}[/tex]

The number of steps she have to take in either direction is [tex]\boxed{72{\text{ steps}}}.[/tex]

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Number system

Keywords: Girl, count, steps, moving escalator, going up, 60 steps, same time, direction, standing still, walking down.