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02 Jul 2025, 08:38

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Given conditions,

total length = 80
speed of the escalator = s
R_a = 2 steps/second (Alice's speed relative to the escalator)
R_b =3 steps/second (Bob's speed relative to the escalator)

Condition 1
Part 1: Alice's Journey
Alice walks upward from the bottom to the top.
Alice's speed relative to the ground = 2+s steps/second
Time taken by Alice to reach the top (T_a) = (80)/(2+s) sec

Number of steps Alice walked on the escalator by the time she reaches the top (n):= 2* (80)/(2+s) sec

Part 2: Bob's Journey
Bob walks downward from the top to the bottom.
Bob's speed relative to the ground =3−s steps/sec
Time taken by Bob to reach the bottom (T_b) = (80)/(3-s) sec

Part 3: Relationship between Alice's and Bob's Times
We are given that Bob takes four times as long to reach the bottom as Alice takes to reach the top:
T_b = 4 * T_a
by simplifying,
2+s=4(3−s)
2+s=12−4s
s+4s=12−2
5s=10
s=2 steps/sec

Calculate n steps Alice walked : n = 40 steps

Condition 2
Time when Alice and Bob Meet (t)
Alice and Bob are moving towards each other.

Alice's speed relative to ground = 2+s=2+2=4 steps/sec
Bob's speed relative to ground = 3−s=3−2=1 step/sec

The effective relative speed at which they close the 80-step gap is the sum of their ground speeds (since they are moving towards each other):
Relative speed = (Alice's speed) + (Bob's speed) = 4+1=5 steps/sec

Time to meet (t) = Total Distance/Relative Speed
= 80/5 =16 sec

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02 Jul 2025, 08:42

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Step 1: We must find the escalator's speed (s).
Alice's effective speed is (2+s). Her time is 80/(2+s).
Bob's effective speed is (3-s). His time is 80/(3-s).
Bob's time is 4 times Alice's time, so: 80/(3-s) = 4 * [80/(2+s)].
Solving this gives 2+s = 4*(3-s), which simplifies to 5s = 10, so s = 2 steps/second.

Step 2: We must find n, the number of steps Alice walks.
Alice's total time to reach the top is T = 80 / (2+s) = 80 / (2+2) = 20 seconds.
The number of steps she walks is her speed * her time: n = 2 steps/sec * 20 sec = 40 steps.

Step 3: Finally, we must find t, the time until they meet.
Their speeds relative to the ground are: Alice = 2+s = 4 steps/sec up; Bob = 3-s = 1 step/sec down.
Since they move towards each other, their combined speed is 4 + 1 = 5 steps/sec.
The time to meet is the distance divided by their combined speed: t = 80 steps / 5 steps/sec = 16 seconds.

Thus,
t = 16
n = 40.

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02 Jul 2025, 08:51

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Give, s= speed of escalator, hence absolute speed of Alice = 2+s and of Bob= 3-s

Ratio time taken by Alice and Bob is equal to the ratio of there speed,

hence (2+s)(3-s)=4, hence s = 2 steps/sec
also. speed of Alice = 4 steps/sec and speed of Bob = 1 step/sec

Now, to meet, Alice and Bob should have travelled a total of 80 steps together in let say t time.
Hence, 80= 4t+1t, therefore t = 16 sec. that's our first answer

Now, for steps travelled by Alice, we know total 80 steps are to covered and since the relative speed of Alice is equal to speed of escalator, exactly half of the total 80 steps .ie., 40 steps will be covered by Alice. hence answer for our second option.

One can go about to calculate the steps travelled by Alice, by calculating total time taken and then multiplying the time with relative speed of Alice.

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02 Jul 2025, 08:53

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Ans: t = 16 , n = 40

The escalator moves upward at a constant rate of s steps per second.

Alice starts from the bottom and walks upward at a constant rate of 2 steps per second relative to the escalator. So, Alice is moving upward with (s+2) steps per second

Bob starts from the top and walks downward at a constant rate of 3 steps per second relative to the escalator. So, Bob is moving downward with (3-s) steps per second

Time taken for Alice to reach top: 80/(s+2)

Time taken for Bob to reach bottom: 80/(3-s)

Given: Bob takes four times as long to reach the bottom as Alice takes to reach the top.

80/(3-s) = 4[80/(s+2)]
solving this will give us s = 2 steps/second from this we get Alice = 4 steps/second and Bob = 1 step/second

t the time in seconds after which Alice and Bob meet
Alice and Bob's relative speed = 4+1 = 5 steps/second (remember this is speed with the escalator's speed)
when they meet = 80/5 = t =16 Seconds

for n the number of steps Alice would have walked on the escalator by the time she reaches the top

Alice moving up with 4 steps/second. So, time taken to reach the top = 80/4 = 20 seconds
in 20 seconds, Alice would take = 20*2 = n = 40 steps

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02 Jul 2025, 08:54

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We have an 80-step escalator moving upward at s steps per second.
For Alice:
- Starts at bottom, walks up at 2 steps/second
- Total upward speed: (s + 2) steps/second
- Time to reach top = 80 ÷ (s + 2)
For Bob:
- Starts at top, walks down at 3 steps/second
- Net downward speed: (3 - s) steps/second A
- Time to reach bottom = 80 ÷ (3 - s)
We're told Bob takes 4 times longer than Alice, so:
80 ÷ (3 - s) = 4 × [80 ÷ (s + 2)]
Solving this equation:
(5 + 2) = 4(3 - s)
s + 2=12 -4s
5s = 10
s = 2 steps/second
Now we can find:
Alice's time to top = 80 ÷ (2 + 2) = 20 seconds
- When they meet, Alice has gone up at 4 steps/second for t seconds
- Bob has gone down from step 80, and is at position (80 - t) after t seconds
- Setting these equal: 4t= 80 - t
- So 5t = 80, meaning t = 16 seconds
For n (steps Alice walks): Alice walks 2 steps/second for 20 seconds = 40 steps
Therefore, t = 16 and n = 40.

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02 Jul 2025, 09:10

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steps is 80
rate of step is s
alice rate 2 and bob is 3
relative speed of A = s+2
and that of B = 3-s
time of B = 4 * time of A
time of A = 80/ (s+2)
time of B = 80/(3-s)
80/(3-s) = 4* 80 /(s+2)
s= 2

time when A & B meet
Pa = (2+s)*t ; 4t
Pb = 80-(3-s)*t ; 80-t
Pa= Pb
4t= 80-t
t=16

time taken by Alice ; 80/ 4 ; 20 sec
speed of alice is 2 ; steps 2*20 ; 40

t is 16, n is 40

Bunuel

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Alice and Bob are on an escalator with 80 steps from bottom to top. The escalator moves upward at a constant rate of s steps per second.

Alice starts from the bottom and walks upward at a constant rate of 2 steps per second relative to the escalator. Bob starts from the top and walks downward at a constant rate of 3 steps per second relative to the escalator.

Bob takes four times as long to reach the bottom as Alice takes to reach the top.

Select for t the time in seconds after which Alice and Bob meet, and select for n the number of steps Alice would have walked on the escalator by the time she reaches the top. Make only two selections, one in each column.

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02 Jul 2025, 09:16

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.............Distance............Speed...............Time
A........... 80 ................... 2+s ................... t
B........... 80 ................... 3-s .................... 4t
From above 2 equations, (2+s)t=(3-s)4t; s=2

.............Distance............Speed...............Time
A........... 80 ................... 4 ................... (80/4=20)
B........... 80 ................... 1. .................... (80/1=80)

Select for t the time in seconds after which Alice and Bob meet, and select for n the number of steps Alice would have walked on the escalator by the time she reaches the top. Make only two selections, one in each column.
A moves 4t steps and B moves 1t steps
4t+1t=80
5t=80; t=80/5=16 seconds

A walks 2 steps per second, so in 20 seconds she walks 2*20=40 steps
n=40

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02 Jul 2025, 09:20

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Given,
distance = 80
speed of Alice = 2+s
speed of Bob = 3-s

As per question
80/(3-s) = 4*80/(2+s)
=> s = 2

time taken to meet = 80/(3+2) = 16sec
Distance travelled by Alice = (2+2)*16 = 64 steps

Answer is t=16 and n=64

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02 Jul 2025, 09:23

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Alice's speed = Va = 2+s/sec
Bob's speed = Vb = 3-s/sec

Same distance but Bob took 4 times longer to finish: Va*t = Vb*4t $\to$ 2+s = 4*(3-s) $\to$ s=2

$\implies$ Alice's speed : 4 steps/sec; Bob's speed : 1 step/sec.

(1) Alice's time to reach the top: 80/4 = 20 seconds. Steps she walks n = 20*2 = 40 steps
(2) Distance between Alice and Bob shortened by 5 steps/second, thus after t = 80/5 = 16 seconds they meet each other.

Answer: t=16 ; n=40

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02 Jul 2025, 09:25

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Bunuel

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We can use the concept of upstream and downstream river to solve this question

Speed of Bob = 3 - s
Speed of Alice = s + 2

80/(3 - s) = 4 * 80 /(s+2)

5s = 10

s = 2

Speed of Bob = 1 step / sec
Speed of Alice = 4 step / sec

Assume that Bob covers a distance of d when they meet

d/1 = 80 - d / 4

d = 16

Hence, they meet after 16 seconds

Number of steps Alice climbs = 1/2 * 80 = 40

Answer

t = 16
n = 40

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02 Jul 2025, 09:29

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Important Points:

80 steps from bottom to top
The escalator moves upward at a constant rate of s steps per second.
Alice
- 2 steps per second
- Starts at bottom
Bob
- 3 steps per second
- Starts at top
Bob takes 4 times as long to bottom as Alice to top.

t = time in seconds after allice and bob meet
N =number of steps. allice takes to get to top.

3(steps per second bob) - S (escalator) = 4[ 2(allice steps) + s)]

3/4 - 4s = 2 + s
3/4= 2+ s + 4s
0.75 - 2= 5s
1.25 = 5s
.25=s

Alice total: 2.25 Steps per second

It is at this point where I think I approached this wrong....

Should 80 have been factored in earlier?
Would love some help/feedback!

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02 Jul 2025, 09:38

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Let's break it down by what is given to us:

Speed of the escalator = S steps/second (moving from bottom to top)
Length of the escalator = 80 steps

Alice:
Speed = 2 steps/second
Since she is moving with the escalator, from bottom to top, her speed becomes 2+S steps/second

Bob:
Speed = 3 steps/second
Since he is moving in the opposite direction of the escalator, from top to bottom, his speed becomes 3-S steps/second

Time taken by Alice to cover 80 steps is $T seconds$
Time taken by Bob to cover 80 steps is $4T seconds$

We are supposed to find the following:

$t$ = Time in seconds that it took for Alice and Bob to meet
$n$ = Number of steps Alice walked while moving up the escalator

First, let's find $t$:

Time to meet can be put in an equation as = $\frac{(Total distance)}{(Relative speed of Alice and Bob)}$

Total distance = 80 steps
Relative speed of Alice and Bob = Speed of Alice + Speed of Bob (Remember, we add their speeds because they are moving in the opposite direction)

Relative speed = 2+S+3-S = 5 steps/second

Putting it in the formula to calculate time, we get, $t$ = 16 seconds

Second, let's find $n$:

We know, for Alice, $T$ = $\frac{80}{(2+S)}$
And for Bob, $4T$ = $\frac{80}{(3-S)}$

Solving these two equations, we get S = 2 steps/second

Notice for Alice, she has to cover a distance of 80 steps, when she herself is walking at 2 steps/second and the escalator is also covering 2 steps/second for her

In other words, for the distance she has to cover, half is being covered by her walking, and the other half is being covered by the escalator
So, of the 80 steps she covered, 40 were covered by her walking

So, $n$ = 40

So the final answer becomes:

$t$ = 16 seconds
$n$ = 40 steps

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02 Jul 2025, 09:39

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lets see
S per sec = speed of esc
S+2 = Actual speed of alice
3-S = Actual speed of Bod

80/(3-S) = 4* 80/(S+2) => this gives S = 2
Time taken by alice = 80/4 = 20 sec. so Steps taken by alice => 20 * 2 => 40 steps so n = 40

Now lets use Relative velocity concept to find out when they meet
rel v = S+2+3-S = 5
t = 80/5 = 16

so n = 40, t = 16

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02 Jul 2025, 09:50

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Alice speed = 2+s
Bob speed=3-s

2+s=4(3-s)
s=2

4t+t=80
t=16

80 steps total already given.

Ans 16 & 80

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02 Jul 2025, 09:53

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Bunuel

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From the given info Alice speed is 2+s, where as Bob's speed is 3-s (As he need to come down s<3).

We also know 4(80/2+s) = 80/3-s => 12-4s =s +2 => s=2.

So the speed of the elevator is 2.

We need to find both meet, Relavtive speed is 4+1 =5, 80/5=> 16.
t=16.

#Steps alice walks, So time it take her to reach top is 80/4 = 20. Her speed is 2, So 40 steps.

n=40, Hence IMO t=16s, n=40 steps

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02 Jul 2025, 09:54

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Bunuel

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80/2+S=320/3-S , s= 1

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Bunuel

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Speed of Bob = 3 - s
Speed of Alice = s + 2

Time = Distance / Speed

Time taken by Bob = 80 / (3 - s)

Time taken by Alice = 80 / (s + 2)

80 / (3-s) = 4 * 80 / (s+2)

Divide by 80 on both sides

1/(3-s) = 4/(s+2)

s + 2 = 12 - 4s

5s = 10

s = 2

Speed of Bob = 1 step / sec
Speed of Alice = 4 steps / sec

Time taken By Bob : Time taken by Alice = Speed of Bob : Speed of Alice

Ratio of speeds = 1 : 4

Hence time taken is in the same ratio = 80 / 5 = 16

As the speed of Alice is same as the speed of escalator the number of steps covered by both are equal. Hence, out of 80 steps, alice covers 40 steps

Answer:

t = 16
n = 40

1 2 3 4 5

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